problem solving maze game

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Spaceship Maze

Can you help the rocket find its way to Earth? Work your way through ten levels of exciting online space mazes. How quickly can you get through them all?

Use the arrow keys or tap the screen to move the rocket through the maze. The goal is to get home to Earth. Keep going until you find the path that will take you there! When you finish one maze, another will follow. Each level gets slightly harder than the last until you beat all ten levels!

These online mazes always provide a new and exciting experience. Even after children have beaten all of the levels once, they can still try again. Each time they play, the mazes for each level will be different. Doing mazes provides many benefits for children. While doing mazes, children develop spatial awareness, practice problem-solving, and learn to strategize.

Online mazes are great for indoor recess, as a center, or for fast finishers. Plus, try our other fun online educational puzzles !

Lots of mazes to try out! Can print a maze to do later.

12 Free Online Mazes

Can you make your way through all these mazes?

Stacy Fisher is an expert on crafting, sewing, and frugal finds, sharing her knowledge of hands-on DIY creations, finding freebies, and budgeting. Stacy was a guest on "The Dr. Oz Show" and featured in Woman's World Magazine. She has nearly two decades of writing and editing experience.

Online mazes are a fun online activity that kids from preschool all the way up to middle school enjoy. They'll have a great time, and you'll know they're improving their problem-solving skills at the same time.

These online mazes are organized into easy, intermediate, and hard. Most of them are presented as maze games, but some are simple online mazes where your mouse acts as the pen.

Prefer to play with paper? Among many other free printable activities for kids are printable mazes , such as ones geared toward Dr. Seuss . Your child might also enjoy other free educational online games like connect the dots games and online color by number games .

Easy Online Mazes

These are easy online mazes that are great for preschool and Kindergarten kids.

Turtle Maze : Guide the turtle down the path to reach the apple.
Cars Maze Game : A simple maze where your toddler will get to guide a car along a street.
Soccer Ball Maze: Take the spinning soccer ball through a maze filled with dangerous ants that will take you back to the start.
Fish Maze : Lead the fish through the bright yellow maze to move onto the next level.
Rocket Maze : Take your rocket to the orange star as fast as you can. Watch out for the spinning planets, though, they'll take you right back to the start.

Intermediate Online Mazes

You'll find these free online mazes get a little tougher and are perfect for kids in grades 1-3.

Spaceship Maze : Use your arrow keys to guide the spaceship through the maze to make it back to earth.
Medium Maze #1 : Use your mouse as a pen to get from the start to the finish of this online maze.
Medium Maze #2 : Another intermediate online maze where you'll need to get to the finish by using your mouse.

Hard Online Mazes

Here are some hard online mazes that kids in grades 4 and up will love.

Maze Speedrun : You have to think ahead, and beat the clock, to get through this maze.
Online/Print Maze : Print or play online, and choose from over a dozen mazes, including invisible mazes and one with lots of twists and turns.
Hard Maze #1 : A very difficult maze where you'll use your mouse as a pen.
Hard Maze #2 : Another very hard online maze. Can you solve it?

Free Play + No Downloads = Win

Play This Game Now

Play Play Maze as a stand alone web app.

Play Maze Puzzle Solving Timed Brain Teaser Maze Game

This is a timed maze game with clean controls and a variety of labyrinth puzzle courses to choose from. Pick any of the 5 preselected mazes shown from the menu. To further challenge yourself, solve a random generated maze based on difficulty levels. Choose between easy, normal, or hard mode. The time is set and varies depending on the complexity and size of the maze.

Select a maze from the main menu.
You can select a random maze based on difficulty level.
Navigate your way through the maze and finish within the time limit.
Finish solving a maze as fast as you can to beat your previous time record.
Players lose if they can’t solve the puzzle level within the time limit.

Play Maze Game Online

Children and adults can play this free online maze puzzle solving game by clicking in the window below.

Alternatively kids and parents can play this timed maze puzzle labyrinth game as a web application here .

Play Maze Game Play Instructions

How to Play

Pick a preselected maze or solve a random maze based on difficulty level. Solve the maze puzzle as fast as you can within the given time limit.

Click on “Select maze” or “random maze” to start.
For “Select maze”, pick from a variety of available mazes. Once you’ve chosen a maze from the 5 preselected courses, click “Select Maze.” Be ready because the timer starts right away.
For “Random Maze”, pick from easy, normal. and hard. Once you select a level, the game will randomly generate a maze for you depending on your preferred difficulty. Be ready because the timer starts right away.
When you tap or click cog button on the upper right, it collapses down to other buttons.
Selecting the sound button allows you to mute or unmute the game’s music.
The icon with opposite arrows is a full screen button. Click it to put your view on full screen or collapse your view.
On mobile or tablet, you need to have a horizontal orientation for the game to work.
On mobile or tablet, play the game as a web application for a better experience. You can find the link here .
To move your circle, you can use the W, A, S, D keys on your keyboard. That’s left for A, right for D, up for W, and down for S.
You may also use the directional arrow keys on your keyboard to navigate through the maze.
Use your finger to press on the circle and guide it toward the direction you want it to go.
On the lower left corner, press the left or right arrow.
On the lower right corner, press the up and down arrow.
Use whichever controls you’re most comfortable with to complete the mazes as fast as you can.
Preselected mazes come with different time limits. Some mazes have a limit of 35 seconds. Harder, more complex labyrinth courses give players 60 seconds to find their way and solve the puzzle.
You have 30 seconds to complete an Easy maze.
You have 40 seconds to complete a Normal maze.
You have 40 seconds or more to complete a Hard maze.
The game is pretty straightforward. You just need to guide the circle toward the finish line.
It’s best if you plan your route. But don’t take too much time. Some passages lead to a dead end and will likely consume a lot of time to correct.
Stay calm and do not panic. The more relaxed you are, the better focus you’ll have in solving the course.
You can pause the game by selecting the cog button on the upper right corner.
This collapses and shows a door button that let’s you exit the game. Click or tap it.
The game will ask you if you want to go back to the opening menu.
Tap or click “Confirm” to go back to the opening menu.
It would be hard to maintain your focus if you need to go to the bathroom or do something else. Make sure to start a maze with uninterrupted time.

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Mobile Friendly Cross Browser Support

This game is rendered in mobile-friendly HTML5, so it offers cross-device gameplay. You can play it on mobile devices like Apple iPhones, Google Android powered cell phones from manufactures like Samsung, tablets like the iPad or Kindle Fire, laptops, and Windows-powered desktop computers. All game files are stored locally in your web browser cache. This game works in Apple Safari, Google Chrome, Microsoft Edge, Mozilla Firefox, Opera and other modern web browsers.

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Plays.org published this Maze on May 25, 2021 / One Comment -->

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One comment on "Play Maze Puzzle Solving Timed Brain Teaser Maze Game"

Play Maze is a much harder set of maze puzzles for the seasoned player. You have a small window to navigate through the predetermined maze. Get to the end point as fast as you can before the timer runs out. You can choose between a predetermined course or a random one. To navigate the maze, touch the screen or use the mouse. You may also use the keyboard A,W,S D, keys or arrow keys to navigate.

Your allotted time limit depends on the difficulty rating of the maze. Easy mazes have a 30-second timer, whereas difficult ones give you 40 seconds or more. Remember, get started as soon as the maze loads. The timer starts early.

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Instructions

Tap the on-screen Arrows to move. Your goal is to make it to the blue square as quickly as possible. Try to complete the maze without taking any extra steps!

Use the Arrow Keys to move. Your goal is to make it to the blue square as quickly as possible. Try to complete the maze without taking any extra steps!

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LESSWRONG LW

Problem solving with mazes and crayon.

I want to talk about a few different approaches to general problem solving (for humans). It turns out that they can all be applied to mazes, so I’ll use some disney-themed mazes to illustrate each approach.

We’ll start off with some traditional path-search algorithms (DFS, BFS, heuristic). Next, we’ll talk about how these algorithms can fall short for everyday problem solving. Then we’ll move on to the interesting part: two techniques which often work better for everyday problem solving, and lend some interesting insights when applied to mazes.

I’ll assume no technical background at all, so if you’ve seen some of this stuff before, feel free to skim it.

DFS and BFS

You have a maze, with a start point and an end point, and you are searching for a path through it. In algorithms classes, this problem is called “path search”.

The very first path search algorithms students typically learn are depth-first search (DFS) and breadth-first search (BFS). Here’s DFS, applied to the Pinocchio maze above:

Basically, the DFS rule is “always take the right-most path which you haven’t already explored”. So, in the Pinoccchio maze, we start out by turning right and running until we hit a wall (the first “x”). Then we turn around and go back, find another right turn, and hit another dead end. Turn around again, continue…

That’s depth-first search. We go as far as we can down one path (“depth-first”) and if we hit a dead end, we turn around, back up, and try another path.

Breadth-first search, on the other hand, tries all paths in parallel:

In this snapshot, we’ve hit three dead ends, and we have four separate “branches” still exploring the maze. It’s like a plant: every time BFS hits an intersection, it splits and goes both ways. It’s “breadth-first”: it explores all paths at once, keeping the search as wide as possible.

There’s lots more to say about these two algorithms, but main point is what they have in common: these are brute-force methods. They don’t really do anything clever, they just crawl through the whole maze until they stumble on a solution. Just trying out paths at random will usually solve a maze about as quickly as DFS or BFS (as long as you keep track of what you’ve already tried).

Let’s be smarter.

Heuristic Search

A human solving a maze usually tries to work their way closer to the end. If we’re not sure which way to go, we take the direction which points more toward the goal.

Formalizing this approach leads to heuristic search algorithms. The best-known of these is A* , but we’ll use the more-human-intuitive “greedy best-first” search. Like breadth-first search, best-first explores multiple paths in parallel. But rather than brute-force searching all the paths, best-first focuses on paths which are closest to the goal “as the bird flies”. Distance from the goal serves as an heuristic to steer the search.

Here’s an example where best-first works very well:

By aggressively exploring the path closest to the goal, Pluto reaches his bone without having to explore most of the maze.

But heuristic search comes with a catch: it’s only as good as the heuristic. If we use straight-line distance from the goal as an heuristic, then heuristic search is going to work well if-and-only-if the solution path is relatively straight. If the solution path wiggles around a lot, especially on a large scale, then heuristic search won’t help much. That’s what happens if we apply best-first to the Pinocchio maze:

… the best-first solution in this case is almost identical to DFS.

General Problem Solving

In AI, path search is used to think about solving problems in general. It turns out all sorts of things can be cast as path search problems: puzzles, planning, and of course navigation. Basically any problem whose solution is a sequence of steps (or can be made a sequence of steps).

What do our maze-solving algorithms look like in a more general problem-solving context?

Brute-force search is, well, brute-force search. It’s trying every possible solution until you hit on one which works. If we use the random variant, it’s randomly trying things out until you hit something which works. I do not recommend solving problems this way unless they are very small, very simple problems.

Heuristic search is more like how most people solve problems, at least in areas we aren’t familiar with. We’ll have some heuristics, and we’ll try stuff which the heuristics suggest will work well.

One common version of heuristic search in real-world problem solving is babble and prune: brainstorm lots of random ideas, then pick out a few which seem promising based on heuristics. Lather, rinse, repeat. This goes by many names; in design, for example, it’s called “flare and focus”. It’s like e-coli path search : flail around, run in a random direction, and if it looks like it’s working, keep going. Otherwise, flail around some more and run in a new direction.

The problem with any heuristic search is that it’s only as good as our heuristic. In particular, most heuristics are “local”: like the straight-line distance heuristic, they’re bad at accounting for large-scale problem structure. Without some knowledge of large-scale problem structure, local heuristics will lead us down many dead ends. Eventually, when we find the “right” solution, we realize in hindsight that we weren’t really solving the right problem, in some intuitive sense — more on that later.

Upping our Game

So we have algorithms which correspond to blindly flailing about (brute force), and we have algorithms which roughly correspond to how humans solve many problems (heuristics). But the real goal is to come with better ways for us humans to solve problems. We need to up our game.

For single-shot problem solving, it’s tough to do better than heuristic path search. One way or another, you have to explore the problem space.

But in the real world, we more often face multiple problems in the same environment. We have jobs, and our jobs are specialized. It’s like needing to find paths between many different pairs of points, but all in the same maze. In this context, we may be able to invest some effort up-front in better understanding the problem space (e.g. the maze) in order to more easily solve problems later on.

(Technical note: here, the usual textbook pathfinding algorithms are not so good. All-pairs path search tries to explicitly represent distances between each pair of points, which is out of the question for real-world high-dimensional problem solving.)

Bottlenecks

Here’s an interesting way to look at the Pinocchio maze:

Note that the highlighted walls have exactly one gap in them. If Pinocchio wants to get from one side of the highlighted walls to the other, then he has to go through that gap.

This observation lets us break the original problem up into two parts. First, Pinocchio needs to get from the start to the gap. Next, he has to get from the gap to the end. During the first half, he can completely ignore all of the maze below the highlight; during the second half, he can completely ignore all of the maze above the highlight.

This is a bottleneck: it’s a subproblem which must be solved in order to solve the main problem.

Once we have identified a bottleneck, we can use it in conjunction with other search algorithms. For instance, rather than running heuristic search on the full maze, we can use heuristic search to get Pinocchio to the gap in the wall, and then run a second heuristic search from the gap to the end. This will do at least as well as the original heuristic search, and usually better.

Even more powerful: unlike search methods, a bottleneck can be re-used for other problems. It’s a property of the problem space, not the problem itself. If Pinocchio is starting anywhere in the top half of the maze, and needs to get anywhere in the bottom half, then the same bottleneck applies. It can even be useful to know about the bottleneck when we don’t need to solve it for the problem at hand; consider this maze:

Having identified the bottleneck, we can see at a glance that the entire bottom half of the maze is irrelevant. If Anna crosses the gap, sooner or later she’ll have to turn around and come back out again in order to reach Elsa. In other words: knowing about a bottleneck you don’t need to solve is useful, because you can avoid it.

Bottlenecks in Real-World Problems

Note that, in order for the bottleneck to add value on top of heuristic search, the bottleneck should be something which the heuristic search would not efficiently solve on its own. For instance, if we’re using a straight-line-distance heuristic in a maze, then a bottleneck is most useful to know about if it’s not near the straight line between start and finish. If there’s a bottleneck we must cross which is way out to the side, then that’s a useful bottleneck to know about.

Another way to word this: the ideal bottleneck is not just a subproblem which must be solved. It’s a maximally difficult subproblem , where difficulty is based on the original heuristics.

This is how we usually recognize bottlenecks in real-world problems. If I’m a programmer trying to speed up some code, then I time each part of the program and then focus on the section which takes longest: the slowest part is the “most difficult” subproblem, the limiting factor for performance. If I want to improve the throughput of a factory, then I look for whichever step currently has the lowest throughput. If I’m designing a tool or a product, then I start by thinking about the most difficult problem which that product needs to solve: once that’s figured out, the rest is easier.

Personally, I find a lot of “shower insights” come this way. I’m thinking about some problem, mulling over the hardest part. I focus in on the hard part, the bottleneck itself, forget about the broader context… and suddenly realize that there’s a really nice solution to the bottleneck in isolation. What’s hard with the original heuristic often becomes easy when we focus on the bottleneck itself, and apply bottleneck-specific heuristics.

In a maze, a bottleneck way to the side is hard using the straight-line distance to the finish as an heuristic. But if we aim for the gap in the wall from the start, then it’s easier.

To find the “right” solution, focus on the right problem. That’s what bottlenecks are all about.

Let’s go back to the Anna and Elsa maze. Here’s an equivalent way to represent the bottleneck we highlighted in that maze:

Half the maze is orange, the other half is green, and the two touch at exactly one point: the green dot, right where we identified a bottleneck. If we want to go from an orange point to another orange point, then we only need to worry about the orange part of the maze. If we want to move between orange and green, then we must cross the green point.

Here, we’ve “chunked” the maze into two pieces, and we can think about those two pieces more-or-less independently. Here’s chunking on the Pinocchio maze, with a few more colors:

For Pinocchio to get to school, he must start in the yellow chunk and get to the orange chunk. We can only get from yellow to orange via green, so the color path must be yellow -> green -> orange (and we can ignore the blue part of the maze entirely). The original maze is broken into three parts, one within each color chunk, and each problem can be solved separately.

(Side question: there’s at least one more-human-intuitive way to apply chunking to mazes. Can you figure it out?)

Once we’ve represented the maze using colored chunks, and once we know how to use that information, we can use it to move between any two points in the maze. “Local” moves, within the same color chunk, can ignore all the other chunks. Larger-scale travel can focus first on the sequence of color moves — essentially a path through a larger-scale, more abstracted maze — and then zoom in to find paths within each chunk. Ideally, we choose the chunks so that an heuristic works well within each piece.

This is actually how humans think about moving around in space. Our mental map of the world contains five or six different levels, each one more granular than the last. When planning a path, we start with the highest-abstraction-level map, and then zoom in.

More generally, experts in a field process information more efficiently by chunking things together into more abstract pieces. Expert chess players don’t think in terms of individual pieces; they think in terms of whole patterns and how they interact. For a human, this is basically what it means to understand something.

I frequently see people tackle problems with tools which resemble heuristic path search — e.g., a brainstorming session, followed by picking out the most promising-sounding ideas.

I think a lot of people just don’t realize that there are other ways to tackle a problem. It’s possible to look at properties of the problem space — like bottlenecks or chunks — before proposing solutions . These are (some of) the pieces from which understanding is built, and when they work, they allow much more efficient and elegant approaches to problems.

Promoted to curated, here are my thoughts:

This post was really easy to get into, was fluently written and generally was engaging to read.
It actually taught me a potential new mental motion, or at least heavily clarified one, in the way it illustrated how bottlenecks in mazes influence the optimal strategy. I've done an undergraduate in CS, and so done lots of path-searching problems, and was familiar with the basic concept, but this connected my abstract idea of identifying bottlenecks with my intuitive planning sense.
Great use of pictures and visual media. I have a guess that we are lacking that on LessWrong, and that a lot of our posts could be made a lot better with more diagrams. And for that I also liked how the diagrams were kinda janky photos of pieces of papers with crayon on them, because it set an example of how you can add diagrams to a post without needing to put in a ton of time and effort in illustrator or something.
I think I would have loved a bit more generalization of the concept of mazes. There is the difficult question of what problems are actually shaped like the maze problem analyzed in this post, and I think the biggest thing preventing me from using the ideas in this post, is a lot of uncertainty about when it is actually a good idea to apply them.
This is good as a stand-alone post, and I think it's valuable to have stand-alone posts, but I am also always in favor of linking to existing discussion of a concept, that we've had on LessWrong. In this case links to the section of the sequences were Eliezer talks about a bunch of graph-search algorithms, and some of the existing writing on chunking would have been great, but I understand that finding those posts can be quite a bit of a pain.

Overall, very happy about this post.

This is great. Just one nitpick

It won't literally do "at least as well as the original heuristic", I'm pretty sure. You could construct a maze where going in the direction of the bottleneck leads you into a longer path than going into the direction of the end. It'll of course do better in expectation.

Yeah, on reflection that's right, I didn't think it through properly.

I like the post. People naturally stick with one of the first few solutions that make sense to them (us), without investing a little more time into investigating the general properties of the problem space. We software designers are very prone to this. A couple more general patterns: working with the duals, working backwards.

Could you give a few examples of what you mean by working with the duals, both in the maze context and otherwise? It brings at least one good maze strategy to mind for me, but the word is used in multiple ways, so I'm curious whether we're thinking of similar things.

Do you perchance know anything about higher-dimensional mazes or labyrinths? As actual games, rather than metaphors for higher-dimensional problem solving.

I do not, but I have always suspected there must be a community of such people somewhere.

No idea. Just off the top of my head, the exponential growth in volume would be an issue for mazes in higher dimensions. Four would probably still be workable, though.

There's this http://miegakure.com/ in development for several years now

A neat example of the failings of heuristics when applied to mazes is the hedge maze at Egeskov Castle, Denmark ( aerial photo , 2D diagram ). It's constructed such that if you choose paths seemingly pointing towards the goal, you'll get caught in an infinite loop - and in fact, the most effective route can be found through a heuristic of choosing the path seemingly pointing away from the goal.

This is really interesting and easy to understand, except I feel like I'm still missing a sense of how to actually apply this to the real world. I think this would really benefit from some examples of how one might apply each strategy to specific real-world problems.

This reply is several years late, but what's a specific real-world problem (tho maybe one closer to a 'toy' problem like the mazes in this post) that you think would serve as a good example for which to (try to) apply the strategies described in the post?

I won't give any spoilers, but I recommend "how to efficiently reach orbit without using a rocket" as a fun exercise. More generally, the goal is to reach orbit in a way which does not have exponentially-large requirements in terms of materials/resources/etc. (Rockets have exponential fuel requirements; see the rocket equation .)

Does "reach orbit" mean put something in orbit or put a human being (and not kill them)? The latter seems pretty hard to do, practically, with current technology, without using rockets (to at least setup an 'efficient' system initially).

(Would you link to some spoilers? I'm curious what you have in mind as solutions!)

Live human being is indeed the harder version. I recommend the easier version first, harder version after.

The latter seems pretty hard to do, practically, with current technology, without using rockets (to at least setup an 'efficient' system initially).

Ah, but what specific bottlenecks make it hard? What are the barriers, and what chunking of the problem do they suggest?

Also: it's totally fine to assume that you can use rockets for setup, and then go back and remove that assumption later if the rocket-based initial setup is itself the main bottleneck to implementation.

This is totally cruel nerd sniping by-the-way :)

Excellent post! Your explanations were interesting and intuitive for me, even though I don't know much of anything about computer science.

Thanks! Glad it made sense, I wasn't sure it would.

I really like the maze analogy! The pictures help a whole bunch in explaining the concepts!

Related: I don't know if this has a direct analog to a CS algorithm, but one other heuristic which seemed like it might have been good to try was drawing a line from both start and finish incrementally to see how they could touch in the middle.

Also not sure if it's a standard concept in path search, but searching from both ends seems fruitful, even in its most naive implementations:

if the graph is sufficiently random, I think it should take the expected fraction of the maze you have to walk from N/2 to sqrt(N)
you can apply it to either BFS or DFS, though I suspect that in practice DFS works better (and even better still if you DFS by switching your first decision (rather than the OP's algorithm of switching your last )
naturally, there's plenty of room for layering heuristics on top, like seeking towards the opposing cursor, or even the nearest point of the opposing explored-so-far surface...

What comes to my mind is a kind of algorithm for auto-chunking: repeatedly take a dead end and fill it in until the point where it branches off from a hallway or intersection. Eventually you're left with a single hallway from entrance to exit.

I'm not certain how it's an improvement over DFS, though.

Answering my own musing: it's implementable as a 2-state cellular automaton, rather than requiring the ~5 states that distributed DFS does. So there's that.

I'm sold on the ideas of heuristics and identifying bottlenecks in solving problems. The concepts are very generally applicable. But I'm not convinced the particular mental motions for solving mazes are the correct ones for solving problems in general.

When I'm solving a math exercise like "Show that the complement of a finite set of k points in R^n is simply-connected if n ≥ 3", I do things like

contemplate the nature of the complement of a finite set of k points in R^n until I can visualize it and correctly answer basic questions about it, like "is it finite or infinite" and "why isn't it simply-connected if n<3"
list all the theorems in the relevant chapter of the topology book, especially the ones that tell why sets are simply-connected (probably I need to use the van Kampen theorem, but there could be others)
think about properties of the set in question that might lead me closer to the hypotheses
repeat steps 2-3 until I develop a general plan that seems likely to work ("Show that the set is homeomorphic to a different set S; partition S into sets that satisfy the hypotheses of the van Kampen theorem; use the van Kampen theorem to show that S is simply-connected; conclude that the original set is simply-connected")
implement the plan

There are definitely heuristics here! But in topology exercises, identifying a bottleneck as severe as a one-empty-space wall is pretty rare, and chunking is usually not possible before the problem is already almost-solved. The core steps in the solution seem to be deconfusion, forward-chaining, and back-chaining, with chunking as an incidental step and identifying a bottleneck as something to happen if you're really lucky.

When I'm answering an open-ended question like "how do I solve alignment", I notice the same thing, with two complications:

There is often no single obvious path, either forwards or backwards. This means we need more deconfusion and better heuristics.
I also need to do much more conscious backchaining than conscious forward-chaining, so there's also an asymmetry between the forward and reverse searches.

But it still seems to be true that bottlenecks are hard to find deliberately. I'm guessing that the examples of bottlenecks-- "programmer trying to speed up some code", or "throughput of a factory", have a special structure that makes bottlenecks easy to find. They are already factored into serial steps, and it's possible to find the bottleneck just by thinking about each of the steps separately.

It sounds like your current strategies are mostly adapted to the sorts of problems found in textbooks or homework assignments - problems which are "easy" in some sense. Such problems are typically already expressed in an ontology which reduces the search-space to something reasonably small, so just listing everything, forward-chaining, and back-chaining will probably be enough. In a large search space in which solutions are rare, those methods do not have the big-O efficiency to work.

That said, many of those methods are still necessary steps even on questions with larger search spaces - for instance, visualizing the main objects in the problem.

I really like this post, it's very informative about the kind of search algorithms that exist. I had one question though, in my experience as a child solving mazes with a crayon I stumbled upon the tactic of starting at the end and getting to the beginning. This worked really well for me, but it always confused me as to why it was easier. Does anyone have a hypothesis as to why this might be?

The mazes may have been constructed to be hard from the front. (Lesser hypotheses: If you switch to the back when you notice it's hard from the front, on average it's going to become easier. If you switch after having explored the maze a little from the front, you already know where to go.)

Also: Searching from the back in addition to searching from the front can be really helpful in reconstructing the overall path! In computer science, we might call this "breadth-first search from both sides" or " bidirectional search " ( image ); the insight being that you keep searching until your two search trees have some point in common. This is nice because you explore fewer total paths -- in a large maze, the number of paths you explore grows exponentially as you get farther away from the starting location.

The analog in general problem solving might be to look both backwards and forwards. If I want to e.g. be happily married with kids in 5 years, I should not only think about my possible actions today ("go on more dates", "reconnect with old crushes"), but also backwards from my destination ("which marriages are happy", "how to raise children"), and look for how the two can connect ("ah, Eve is an old friend from high school who shares this interest in child rearing").

This is also a technique in math proofs. From XKCD :

"Handy exam trick: when you know the answer but not the correct derivation, derive blindly forward from the givens and backward from the answer, and join the chains once the equations start looking similar. Sometimes the graders don't notice the seam."

Except: sometimes, instead of tricking the grader, you actually just find the correct derivation!

I know someone who successfully used that "handy exam trick" in the International Mathematical Olympiad.

(Of course this isn't relevant to the actual discussion, but it might amuse you as it amuses me. Further details that I find pleasing: 1. He took care to make sure that the "seam" lay between the end of one page of working and the start of the next. 2. The people marking the script claimed to have found an entirely different error which was not in fact an error, but of course the country's team leaders didn't send it back and complain because they knew that if they did the much more serious error in logic was likely to be noticed. For the avoidance of doubt, the person in question was not me.)

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Fun Maze is an engaging online maze game for kids. It is divided into three levels- easy, medium and hard. Kids will enjoy hours of endless fun as they explore the different mazes and try to find the correct route to solve the puzzle. Through visual discrimination and eye-hand coordination kids will enhance their problem solving ability. This game will also help kids improve their logical thinking skills, as they play along.

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The maze, with small group walking through a series of spots to navigate a mystery maze.

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Step-by-step instructions.

Place a series of gym-spots or carpet tiles on the floor arranged in several rows and columns.
Starting at one end of the spots, instruct your group to navigate their way through the maze – one step at a time – to identify the secret pathway.
Announce that the path will not travel in a straight line, it will never cross itself, and will not use any spot twice.
Only one person is permitted inside the maze at any point in time.
When a spot is stepped on with two feet, you (as instructor) will indicate if it is part of the secret pathway or not.
Apply an appropriate penalty for each step made in error.
Challenge your group to identify the secret pathway in as few errors/steps as possible.

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It also demonstrates that one is not stuck to being either a follower or leader but it can be a choice we make to be what is needed, either follower, observer or leader and that each role can learn from our experiences and is of importance towards reaching a common goal.

Using the silent version variant is also fantastic as other forms of assistance or support or communication can emerge and this too can produce some wonderful conversations and thoughts about how the group adapts to meet challenges.

David, you point to a number of really useful observations to make with a group when you invite them to reflect on their experience. There is so much more to leadership than just being the person with the loudest voice 🙂

Love the maze… so useful in demonstrations concepts around leadership and learning from experience. This is a go to with groups when exploring the themes around leadership and what leadership looks like and the styles available.

It also allows the group to look and value the role of the follower as well which at times is not as valued as much.

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Strategies and Techniques for Solving Mazes

Mazes can be intriguing puzzles that challenge our problem-solving skills. Navigating through a labyrinthine network of paths and dead ends can be both exciting and frustrating. However, with the right strategies and techniques, solving mazes becomes an enjoyable endeavor. In this article, we will explore some effective approaches to tackle mazes successfully.

1. Start with an Overview

Before delving into the maze, it’s helpful to take a moment to observe and understand the maze’s structure. Look for patterns, openings, and dead ends. Identify the entrance and exit points. This bird’s-eye view will give you a better understanding of the maze’s layout and potential paths to explore.

2. Follow the Right-Hand Rule

One popular technique for solving mazes is the “Right-Hand Rule.” As the name suggests, you keep your right hand touching the wall of the maze at all times. By consistently following the right wall, you can ensure that you explore the entire maze without retracing your steps. This technique works well for mazes with only one possible solution.

3. Employ the Left-Hand Rule

Similar to the Right-Hand Rule, the “Left-Hand Rule” can also be effective. Instead of touching the right wall, you keep your left hand in contact with the maze’s wall. By doing so, you can still navigate the entire maze, but in the opposite direction. The Left-Hand Rule can be particularly useful if you encounter a maze with multiple solutions.

4. Divide and Conquer

If the maze is large and complex, consider dividing it into smaller sections. Focus on solving one section at a time. By breaking down the problem into manageable parts, you can reduce the complexity and increase your chances of finding the correct path. Once you solve each section individually, you can connect them to complete the maze.

5. Look for Patterns

Mazes often feature recurring patterns, especially in their dead ends or twists and turns. By identifying these patterns, you can anticipate where potential dead ends might lie and avoid wasting time exploring fruitless paths. Pay attention to walls or paths that repeat or exhibit symmetrical patterns, as they can provide valuable hints for finding the correct route.

6. Backtrack Strategically

When you encounter a dead end or realize you have taken a wrong turn, don’t be discouraged. Backtracking can be an essential part of maze-solving. However, it’s crucial to backtrack strategically. Revisit the last intersection or choice point and explore an alternative path. By systematically revisiting previous decisions, you increase your chances of finding the correct route.

7. Utilize Trial and Error

Sometimes, solving a maze requires experimentation and trial and error. If you encounter a particularly challenging section, don’t hesitate to try different paths to see where they lead. Mark your progress using a pencil or mental notes to keep track of the paths you have explored. Even if some attempts lead to dead ends, they provide valuable information and contribute to your overall understanding of the maze.

8. Don’t Give Up

Solving mazes requires a combination of logical thinking, observation, and perseverance. By employing strategies like the Right-Hand Rule or Left-Hand Rule, dividing the maze, looking for patterns, and using trial and error, you can tackle even the most intricate mazes with confidence. So, the next time you face a maze, embrace the challenge and embark on an exciting journey to find the way out!

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Jr. Music Maze Grab an instrument and lay down a fresh beat with your friends as you explore this mini-maze.

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Continuous Finger Maze Send your finger on a winding path around the bend and back.

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Infinity Mirrors How many yous can you count?

Funhouse Mirrors Play with angles of reflection to distort reality in wacky ways.

Prism Table Mix, reflect, and refract light to learn how our brains perceive the colors of the rainbow.

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Which Creature is Bigger You can’t always believe your eyes. Use tools to help your brain make sense of what you think you see.

Seeing Colors Can you trick your brain into turning black and white into a blur of color?

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Front Aging Neurosci

Development and Evaluation of Maze-Like Puzzle Games to Assess Cognitive and Motor Function in Aging and Neurodegenerative Diseases

1 Gerontechnology & Rehabilitation, University of Bern, Bern, Switzerland

2 ARTORG Center for Biomedical Engineering Research, University of Bern, Bern, Switzerland

Alvin Chesham

Narayan schütz, angela amira botros, tim vanbellingen.

3 Neurocenter, Luzerner Kantonsspital, Lucerne, Switzerland

Jean-Marc Burgunder

4 Neurozentrum Siloah and Department of Neurology, Swiss Huntington’s Disease Center, University of Bern, Bern, Switzerland

Julia Müllner

5 Department of Neurology, University Neurorehabilitation, Inselspital, Bern, Switzerland

René Martin Müri

Prabitha urwyler, associated data.

Data pertaining to the obtained results may be provided upon request.

There is currently a need for engaging, user-friendly, and repeatable tasks for assessment of cognitive and motor function in aging and neurodegenerative diseases. This study evaluated the feasibility of a maze-like Numberlink puzzle game in assessing differences in game-based measures of cognition and motor function due to age and neurodegenerative diseases. Fifty-five participants, including young (18–31 years, n = 18), older (64–79 years, n = 14), and oldest adults (86–98 years, n = 14), and patients with Parkinson’s (59–76 years, n = 4) and Huntington’s disease (HD; 35–66 years, n = 5) played different difficulty levels of the Numberlink puzzle game and completed usability questionnaires and tests for psychomotor, attentional, visuospatial, and constructional and executive function. Analyses of Numberlink game-based cognitive (solving time and errors) and motor [mean velocity and movement direction changes (MDC)] performance metrics revealed statistically significant differences between age groups and between patients with HD and older adults. However, patients with Parkinson’s disease (PD) did not differ from older adults. Correlational analyses showed significant associations between game-based performance and movement metrics and performance on neuropsychological tests for psychomotor, attentional, visuospatial, and constructional and executive function. Furthermore, varying characteristics of the Numberlink puzzle game succeeded in creating graded difficulty levels. Findings from this study support recent suggestions that data from a maze-like puzzle game provide potential “digital biomarkers” to assess changes in psychomotor, visuoconstructional, and executive function related to aging and neurodegeneration. In particular, game-based movement measures from the maze-like puzzle Numberlink games are promising as a tool to monitor the progression of motor impairment in neurodegenerative diseases. Further studies are needed to more comprehensively establish the cognitive validity and test–retest reliability of using Numberlink puzzles as a valid cognitive assessment tool.

Introduction

Neurodegenerative diseases comprise a heterogeneous group of age-related disorders with progressive loss of neuronal structure and function (Cummings and Pillai, 2016 ; Kovacs, 2018 ). Neurodegeneration can occur in normal aging and age-related neurodegenerative diseases that include Alzheimer’s disease (AD), Parkinson’s disease (PD), and Huntington’s disease (HD). Affecting different brain areas, patients with neurodegenerative diseases suffer from cognitive, motor, and neuropsychiatric impairments (Cummings and Pillai, 2016 ; Hussain et al., 2018 ; Kovacs, 2018 ). Cognitive functions commonly affected in neurodegenerative diseases include attentional, visuospatial, and constructional, executive, and motor function that detrimentally impact everyday activities and quality of life in normal cognitive aging (Deary et al., 2009 ; Diesing and Rizzo, 2018 ), AD (Rycroft and Giovannetti, 2017 ), PD (Ding et al., 2015 ; Pal et al., 2018 ), and HD (Dumas et al., 2013 ; Paulsen and Long, 2014 ). Strikingly, these cognitive changes vary considerably both between and within individuals in normal aging (Agrigoroaei and Lachman, 2011 ; Mosti et al., 2019 ) and patients with neurodegenerative disorders (Boller et al., 2013 ; White et al., 2018 ; Greenland et al., 2019 ).

Given this variability, it is important to differentiate normal cognitive aging from neurodegenerative diseases in order to detect cognitive deficits early and provide adequate treatment to attenuate further neurodegeneration and cognitive dysfunction (Massaldjieva, 2018 ). However, existing neuropsychological assessments have been criticized for their: (a) unsuitability for repeated use because of time demand and practice effects (i.e., test performance improvements when tests are taken repeatedly); (b) limited sensitivity to detect early and subtle cognitive changes because of ceiling and flooring effects (i.e., high rates of highest or lowest possible test scores); and (c) lack of ecological validity as many cognitive tasks do not reflect cognitive demands of everyday activities (Silverberg et al., 2011 ; Allard et al., 2014 ; Zygouris and Tsolaki, 2015 ; Valdes et al., 2016 ; Howieson, 2019 ).

A novel approach to address these issues is the use of video games to study age and neurodegenerative disease-related differences in cognitive function (Boot, 2015 ; Koo and Vizer, 2019 ). Video games involve solving cognitive challenges that require a range of cognitive abilities and often share properties with psychological tests (Holmgard et al., 2016 ). Video games provide a bulk of performance measures (e.g., reaction and solving time, number of errors and others) that can be used to assess cognitive functioning (Areàn et al., 2016 ; Mandryk and Birk, 2019 ). Furthermore, video games promote prolonged motivation and engagement, and offer performance-based adjustments of task difficulty that can help avoid practice effects with repeated use and accommodate different levels of cognitive ability (Holmgard et al., 2016 ; Diesing and Rizzo, 2018 ; Levy et al., 2018 ). Also, video games involve complex cognitive skills that better reflect everyday cognitive function than simple cognitive tasks (Boot et al., 2013 ). To date, several studies demonstrated associations between video game performance measures and performance on specific cognitive tests, suggesting that video games measure relevant cognitive abilities (Baniqued et al., 2013 ; Oei and Patterson, 2013 ; Martinovic et al., 2015 ) and are feasible to assess cognitive function in older adults (Thompson et al., 2012 ; Tong et al., 2016 ).

With regard to detecting age-related changes in cognitive and motor functioning, maze tasks represent a particularly promising addition to standardized cognitive assessments (de Souza et al., 2013 ). Mazes are complex visual-motor planning and problem-solving tasks that require finding a path from the start to the end of a maze as quickly as possible. Mazes are non-verbal, simple to understand and use, relatively independent of educational level and suitable for a wide range of older adults and persons with cognitive impairment (Marhasev et al., 2009 ; Silva et al., 2017 ). Maze tasks require an interplay between both cognitive and motor processes and are similar to complex everyday activities that require planning and problem solving (Lewis and Miller, 2007 ; Marhasev et al., 2009 ; Howieson, 2019 ). Solving mazes requires multiple cognitive processes that include attentional, visuospatial and visuoconstructional, and executive function (planning, foresight and problem solving) as well as visuomotor function (Snellgrove, 2005 ; Kirsch et al., 2006 ; Carlozzi, 2011 ; Zhao and Marquez, 2013 ).

To date, maze tests have been used in some studies to assess cognitive and motor functioning and demonstrated sensitivity in differentiating healthy normal aging from either mild cognitive impairment (Zhang et al., 2007 ; de Souza et al., 2013 ), Alzheimer’s dementia (Mack and Patterson, 1995 ), PD (Mimura et al., 2006 ), and HD (Deckel and Duffy, 2000 ; Montoya et al., 2006 ). Furthermore, maze task performance was shown as a strong predictor of everyday functioning ability in older adults (Mack and Patterson, 1995 ; Ott et al., 2003 ; Lewis and Miller, 2007 ; Staplin et al., 2013 ).

In the current study, a maze-like Numberlink (NL) puzzle video game adapted to assess cognitive and motor ability in aging and neurodegenerative diseases is presented. NL was first published as a newspaper puzzle column by Sam Loyd (Loyd, 1897 ) and later popularized in Japan as pastime puzzle books (Dudeney, 1917 ; Nikoli, 1989 ; Yoshinaka et al., 2012 ; Adcock et al., 2015 ). In 2012, NL puzzles have been released as a puzzle game for mobile applications called Flow Free ® (Big Duck Games LLC) that has been downloaded over 250 million times and is among the most popular puzzle games (Newman and Newman, 2012 ). In a previous playtest study in older adults, we found that the NL puzzles were particularly enjoyed and rated to meet game characteristics for cognitive tasks and trainings (Chesham et al., 2017 ).

Similar to video games, mazes can be modified in difficulty to match the test-takers’ level of cognitive ability. A number of recent studies have further provided maze tasks with graded levels of difficulty accomplished by varying variables of the maze (McClendon, 2001 ; Ott et al., 2003 ; Blatter et al., 2005 ; Davis et al., 2014 ; Pasek, 2016 ). Variable maze difficulty levels can help prevent practice effects during repeated administration and reduce ceiling and flooring effects by continuously matching the task difficulty to the participant’s cognitive ability level (Davis et al., 2014 ; Loe and Rust, 2019 ). Graded difficulty levels for the NL puzzle game were generated by manipulating game characteristics. This benefits future studies to create game-based adaptive computerized cognitive assessments.

The first aim of this study was to examine the acceptance and usability of a game-based NL task in young, older, and oldest adults and persons with neurodegenerative diseases. In line with recent suggestions, we hypothesize that the use of playful elements of a puzzle video game is user-friendly and enjoyable for older people and patients with neurodegenerative diseases (Polzer and Gewald, 2019 ). The second aim of this study was to examine age- and NDD related differences in NL game-based cognitive (solving time and accuracy) and motor (movement velocity and direction changes) performance.

Following previous findings using maze tasks, our study tests the hypothesis that NL game-based performance measures can be used to distinguish between age groups (de Souza et al., 2013 ) and between healthy controls and patients with HD (Montoya et al., 2006 ) and PD (Mimura et al., 2006 ). Based on our hypothesis that NL puzzles are similar to maze tasks, we propose that NL game-based performance measures are associated with performance on the Snellgrove Maze Task (SMT) and standard measures of visuomotor, visuospatial and constructional, executive, and global cognitive function (Snellgrove, 2005 ; Yew et al., 2011 ). Finally, we assessed whether NL puzzle difficulty can be varied by manipulating game parameters. Following a previous study, we hypothesize that the difficulty of NL puzzles increases with set size and the number of paths (van Kreveld et al., 2015 ).

Materials and Methods

Participants.

In total, 55 participants (36 females) between the ages of 18 and 98 years participated in this study (see Table 1 ). Five groups of participants were enrolled for this study: (1) young adult university students (YA; n = 18, 12 female, mean age 21.83, range 18–31 years); (2) older adults (OA; n = 14, 8 female, mean age 71.14, range 64–79 years); (3) oldest adults (OOA; n = 14, 12 female; mean age 89.93, range 86–98 years) living independently in senior residence apartments. The group of participants with neurodegenerative disease included; (4) patients with HD ( n = 5, 2 female; mean age 49.40, range 35–66 years; Movement Disorder Society – Unified Parkinson’s Disease Rating Scale (MDS-UPDRS-III) 45.75 ± 10.78, range 30–54; Total Functional Capacity 8.0 ± 3.16, range 6–12; Functional Assessment: 15.75 ± 6.18, range 7–21; Independence Scale 81.25 ± 21.75%, range 60%–100%); and (5) patients with PD ( n = 4, 2 female; mean age 67.50, range 59–76 years; MDS-UPDRS-III 17.75 ± 5.38, range 11.00–23.00; disease duration 9.88 ± 3.01 years, range 7.00–14.00 years). HD and PD patients were recruited from the Swiss HD Center, Neurozentrum Siloah; Department of Neurology, Bern, Switzerland; and the Neurology and Neurorehabilitation Center, Luzerner Kantonsspital, Luzern, Switzerland. All HD patients were previously given a clinical confirmed diagnosis of HD. One of five HD patients did not receive any medication, while the rest of HD patients were under medications. The medications were personalized depending on the individual symptoms and covered a wide range (Zolpidem 10 mg 1–2, Risperidone 0.5 mg, Trittico 100 mg, Citalopram 20 mg, Quetiapine 25 mg, Zoldorm 10 mg, Aripiprazole 5 mg, Eltroxin LF 0.05 mg, and Rosuvastatin 10 mg). Patients with PD were diagnosed with PD as per UK Parkinson’s Disease Brain Bank criteria and prescribed anti-Parkinsonian medication (levodopa-dose 412.5 ± 188.75 mg, range 150–600 mg). Three of the PD patients were also prescribed additional drugs depending on their symptoms (Sifrol ER 1.5 mg, Pramipexole 3 mg, Roprinirole 8 mg, and Safinamide 50 mg). All PD patients were in “ON” phase of the medication cycle when participating in this study.

Demographics, clinical and neuropsychological measures, acceptance, and usability by group.

All values, mean ± SD or n (%), p value < 0.05 is significant. YA, Young Adults; OA, Older Adults; OOA, Old oldest Adults; PD, Parkinson’s Disease; HD, Huntington’s disease; TMT, Trail Making Test; SMT, Snellgrove Maze Task; MoCA, Montreal Cognitive Assessment (Score: 1–30), P Perception of Game Training Questionnaire (Rating: 1–7); SUS, System Usability Scale (Score: 0–100), Q Unified Parkinson’s Disease Rating Scale, Z comparison across all groups df = 4, A YA vs. OA, B YA vs. OOA, C YA vs. PD, D YA vs. HD, E OA vs. OOA, F OA vs. PD, G OA vs. HD, H OOA vs. PD, I OOA vs. HD, J PD vs. HD .

Exclusion criteria for participation were insufficient coordinative, motor, and perceptual ability to handle a tablet-computer and history of any additional neurological or psychiatric deficits. All participants had normal or corrected-to-normal vision. Written informed consent was provided in accordance with the latest version of the Declaration of Helsinki prior to participation. The cantonal ethics committees of Bern, Northwest and Central Switzerland, Switzerland granted the ethical approval for this study (2016-01281).

Neuropsychological Assessment

Global cognitive, attentional, visuospatial and visuoconstructional, and executive function was evaluated in all participants. Global cognitive ability was examined with the Montreal Cognitive Assessment (MoCA) that evaluates executive, attentional and visuospatial function, memory, and language (Nasreddine et al., 2005 ). Patients with HD completed the Mini Mental State Examination (MMSE; Folstein et al., 1975 ) instead of the MoCA. MMSE scores for the HD patients were converted to MoCA scores using conversion guidelines from Roalf et al. ( 2013 ). Attention and executive functions were further assessed with the Trail Making Test part A (TMT-A) and part B (TMT-B; Schretlen et al., 1996 ). The TMT-A tests selective attention, visual scanning, and visuomotor processing, and the TMT-B tests divided attention and executive control (Schretlen et al., 1996 ; Strauss et al., 2006 ). Finally, the SMT was used as a screening for multiple cognitive functions that include attention, visuoconstructional ability, and executive functions of planning and foresight (Snellgrove, 2005 ).

Acceptance and Usability Assessment

Subjective acceptance of the NL puzzle game was assessed with the Perception of Game Training Questionnaire (Boot et al., 2013 ). In this questionnaire, participants rated the extent to which they found playing the mazes “enjoyable,” “challenging,” “frustrating,” as well as their motivation while playing the mazes on a seven-point Likert scale. The 10-item System Usability Scale (SUS) was used to measure user experience, usability, and learnability of the NL puzzle game. The SUS provides a composite score from 0 to 100 where a higher number indicates a higher usability (Brooke, 1996 ).

NL Puzzle Task

Task description.

NL puzzles are maze-like link puzzles that involve finding multiple distinct paths to connect pairs of identical objects using non-intersecting and continuous lines (Yew et al., 2012 ). NL puzzles are played on a grid-based (width × height) puzzle board of cells. Some cells contain colored circles with numbers or letters (“dots”) that represent start and end points of paths, while the rest of the cells are empty (Kalvelagen, 2017 ). Dots always come in pairs that have the same color and letter (Hartmann, 2018 ; see Figure 1 , top row). The overall goal of NL puzzles is to connect all pairs of dots with non-intersecting and continuous lines (called “path” or “flow”) such that finally each empty cell in the grid is part of a path (Adcock et al., 2015 ). The following rules must be observed to solve NL puzzles: first, all pairs of identical dots must be connected with single continuous paths. Second, paths can be drawn only in horizontal or vertical direction, must go at least through one empty cell, and cannot go through an empty cell twice. Third, paths are not allowed to cross cells containing dots or intersect with other paths, as crossed paths will be overwritten. Fourth and final, all empty cells must be filled with paths once all pairs of dots are connected (see Figure 1 , bottom row; Newman and Newman, 2012 ; Yew et al., 2012 ; Yoshinaka et al., 2012 ; Kalvelagen, 2017 ; Hartmann, 2018 ; Laurentiz, 2018 ). The main challenge of NL puzzles lies in completing the puzzle in as little time as possible using a minimum number of moves. Moves are counted from the moment a dot is tapped and dragged to draw a path until it is released again. Moves can result in either complete or incomplete paths between two identical dots. Paths can be deleted completely by touching the start or end dot or can be broken up at the point where the path is touched. Hence, to solve a NL puzzle with an optimal number of moves, each pair of dots should be connected exactly once; that is, the number of moves should be equal to the number of pairs of dots (Newman and Newman, 2012 ; van Kreveld et al., 2015 ).

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Numberlink puzzle difficulty levels. The manipulated variables include the width and height of the board, the number of paths, and the total number of turns. The top row depicts the initial, and the bottom row, the solved NL puzzles. The difficulty levels shown from left to right are ( width, height, paths, turns ) = (4, 4, 4, 4), (5, 5, 5, 5), (6, 6, 6, 6), (7, 7, 7, 7), (8, 8, 8, 8).

Comparison Between Maze Tasks and NL Puzzles

NL puzzle games share similarities with perceptual maze tasks such as paper-based and computerized versions of the Porteus Maze Tests and Wechsler Mazes (Porteus, 1945 ; Wechsler, 1949 ; Ott et al., 2003 ; Blatter et al., 2005 ), the Elithorn Perceptual Maze Test (Elithorn, 1955 ; Loe and Rust, 2019 ), and the SMT (Snellgrove, 2005 ). Maze tasks are composed of two-dimensional grids of cells and are made up of a complex set of branching paths (“arms”) confined by fixed walls between cells. The start (“entry”) and end (“exit”) points are located on the outer edge of the maze ( Figure 2 , Maze Task, left). Arms inside the maze are connected by intersections and are either open or closed (“dead-ends”). The goal of maze tasks is to draw a single continuous path from entry to the exit ( Figure 2 , Maze Task, right). This requires a multistep solution composed of a set of connected, directed arms within the maze. Therefore, solving a maze involves selecting multiple, consecutive correct arms and avoiding dead-ends at every intersection of the maze (Bagnall and Zatuchna, 2005 ; Blatter et al., 2005 ; Carlozzi, 2011 ; Pasek, 2016 ).

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Maze Task (A) : maze with cells separated by boundaries. The maze comprises connected paths that are open or closed (dead-ends). To solve the maze, single continuous path (i.e., sequence of connected paths) between start and end must be found. Numberlink Puzzle (B) : Numberlink puzzle consisting of a two-dimensional grid with multiple start and end points (i.e., pairs of identical dots). To solve the puzzle, all pairs of dots must be connected with continuous non-intersecting lines and all empty cells must be part of a path.

Like perceptual mazes, NL puzzles are played on a two-dimensional grid of cells. Instead of finding a single path between a single entry and exit point, NL puzzles involve connecting multiple unordered start and end points (i.e., pairs of identical dots) with multiple continuous, non-intersecting paths ( Figure 2 , NL Puzzle). In addition, unlike mazes, paths in NL puzzles are not delimited by static boundaries, but “pre-determined” by constraints given by the rules of the game (Laurentiz, 2018 ). For this reason, NL puzzles are considered a set of undirected paths that connect multiple start and end points (Yew et al., 2011 ; Alviano et al., 2013 ; Hartmann, 2018 ).

NL Difficulty Level Generation

The NL puzzle game used in this study was adapted according to the Flow Free ® App (Big Duck Games LLC; Newman and Newman, 2012 ). In the Flow Free ® App, difficulty is governed by the size of the grid-based board with sizes ranging from 5 × 5 to 15 × 15. A recent study showed that four game characteristics can be used to model and control the difficulty level of NL puzzles: (1) the size of the board (width × height); (2) the number of paths or pairs of dots; (3) the total number of turns, that is, direction changes of paths to the left or right; and (4) the average city block distance between start and end points of paths (Mutser, 2014 ; van Kreveld et al., 2015 ).

To create fine-grained and controlled difficulty levels for the NL puzzle in this study, a set of 25 board sizes was specified by all combinations of board widths and heights varied over a range from 4 to 8: ( width, height ) = (4, 5, 6, 7, 8) × (4, 5, 6, 7, 8). For each of the 25 board sizes, 1 Mio. random NL puzzles were computer-generated using an open-source Numberlink Generator (Ahle, 2017 ). Following a recent study on determinants of NL puzzle difficulty, the generated puzzles (text files) were computer-processed and sorted by the number of paths and the total number of turns (Mutser, 2014 ; van Kreveld et al., 2015 ). In order to reduce the number of generated NL puzzles for this study, the number of paths and the number of total turns per maze were set between 4 and the maximum value of the width or height of the board. This resulted in a total number of 361 difficulty levels: ( width, height, paths, turns ) = (4, 5, 6, 7, 8) × (4, 5, 6, 7, 8) × [4 ≤ paths ≤ max (width, height)] × [4 ≤ turns ≤ max(width, height)]. Then, two parallel versions of difficulty levels (see Supplementary Material, Appendix ) were created, by parallelizing board size and number of paths ( width, height, paths ). Both sets contained all 15 square ( width = height ) NL difficulty levels ( width, height, paths ) = (4, 4, 4), (5, 5, 4), (5, 5, 5), (6, 6, 4), (6, 6, 5), (6, 6, 6), (7, 7, 4), (7, 7, 5), (7, 7, 6), (7, 7, 7), (8, 8, 4), (8, 8, 5), (8, 8, 6), (8, 8, 7), (8, 8, 8). The non-square ( width ≢ height ) NL difficulty levels were assigned in a parallelized fashion ( Set A ) = (5, 4, 4), (4, 5, 5), …, (8, 7, 8), ( Set B ) = (4, 5, 4), (5, 4, 5), …, (7, 8, 8) but still contain all board sizes. Finally, for both sets, one instance of total number of turns (ranging from 4 to 8) was selected randomly. For both the full set A and B, three sets of difficulty levels were created (see Supplementary Material, Appendix ): a short version ( width, height, paths, turns ) = (4, 5, 6) × (4, 5, 6) × [4 ≤ p ≤ max(width, height)] × [4 ≤ t ≤ max(w, h)] = 12 levels, a medium version ( width, height, paths, turns ) = (4, 5, 6, 7) × (4, 5, 6, 7) × [4 ≤ paths ≤ max(width, height)] × [4 ≤ turns ≤ max(width, height)] = 24 levels, and a long version ( width, height, paths, turns ) = (4, 5, 6, 7, 8) × (4, 5, 6, 7, 8) × [4 ≤ paths ≤ max(width, height)] × [4 ≤ turns ≤ max(width, height)] = 40 levels. The reasoning behind creating short, medium, and long versions of NL difficulty levels was based on preliminary experience and mainly due to time considerations and to avoid overburdening participants.

Data Preparation, Maze Performance, and Movement Metrics

NL puzzle performance metrics were calculated based on the participants’ completion time, accuracy, and touchscreen interaction (“movement”). For each played maze difficulty level, two data files were stored as text files: (1) time-stamped “screenshots” (two-dimensional arrays) representing every change of state in empty cells from between the initial and solved NL puzzle; and (2) raw touch input data from the tablet-computer consisting of time-stamped x and y coordinates of touch points.

First, three response time-based NL performance measures were calculated: total solving time (TST), motor execution time (MET), and mental planning time (MPT). TST was defined as the time elapsed from the initial touch to the solved NL puzzle (see Figure 1 ). Using the touch input data, single moves (“drag movements”) were defined as a sequence of touch points that fall between a touch down and release event (see Figure 3 ). For every move, move duration (i.e., the time between touch down and touch release) was calculated (Antal and Szabó, 2016 ). MET was computed by summing the durations of all moves that were required to complete the NL puzzle. Finally, MPT was calculated by subtracting MET from TST.

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Numberlink motor performance based on tablet touch interaction data. Examples are shown for a young adult (top row), oldest adult (middle row), and patient with Huntington’s disease (HD; bottom row). Touch interaction data are shown for NL puzzle levels solved without errors (false paths) for difficulty levels (width, height, paths, turns) = (4, 4, 4, 4), (5, 5, 5, 5), (6, 4, 5, 6). Green circles indicate the start and red circles indicate the end of a move. Red squares represent turns in the path as detected using the Ramer–Douglas–Peucker algorithm.

Second, two accuracy-based NL performance measures were calculated: the number of false moves and excess moves. False moves were defined as complete or incomplete paths that were not part of the solved maze or, although correct, were deleted before the maze was finally solved. Excess moves were calculated by subtracting the number of paths (i.e., pairs of dots) from the total (i.e., false and correct) number of moves.

Third, movement metrics were computed from the touch data. For this, moves connecting start and end points were extracted. In NL puzzles, moves follow either straight or winding paths. Winding paths change direction from segment to segment depending on the number of turns of the path (see Figures 1 , ,2). 2 ). Because movements in NL puzzles are constrained to vertical and horizontal direction, moves for paths with changing directions (i.e., paths involving turns) were segmented into sub-paths based on the detected path turns (see red squares for detected path turns and red arrows for sub-paths in Figure 3 ) using the Ramer–Douglas–Peucker algorithm (Ramer, 1972 ; Douglas and Peucker, 1973 ; Bleier, 2011 ). For the thus extracted sub-paths and paths without turns (straight), two quantitative movement metrics were calculated: mean velocity (MVE) and movement direction changes (MDC; MacKenzie et al., 2001 ).

MVE represents the average movement speed. MDC represents the number of direction changes within the axis of the detected sub-paths during drag movements. More recently, these measures have been shown successful in differentiating between groups with and without motor impairment (Keates and Trewin, 2005 ; Keates et al., 2005 ; Mertens et al., 2012 ; Montague et al., 2014 ; Papatheodorou et al., 2019 ).

The study consisted of a neuropsychological assessment and a NL puzzle playing session. After written informed consent was obtained, baseline measures of general cognitive ability (MoCA), visual search and visuomotor processing (TMT-A), cognitive flexibility, divided attention, working memory and inhibition (TMT-B), and visuoconstructional ability, planning, and foresight (SMT) were collected.

Next, participants completed a practice block of three NL puzzles: ( w, h, p, t ) = (4, 4, 4, 0), (4, 4, 4, 4), (5, 5, 5, 5) on a tablet-computer (Apple© 12.9” iPad Pro, Apple Inc., Cupertino, CA, United States). Instructions were given that all pairs of same-colored dots must be connected using tap and drag movements to cover all empty cells in the board and that connections will be severed if they intersect. Also, participants were shown the hint button that connects two dots when pressed and encouraged to use a hint should they struggle to solve a puzzle. Thereafter, participants were assigned one of three sets of NL puzzle difficulty levels based on their individual MoCA score and TMT B performance: short version (MoCA < 23, TMT B > 120 s), medium version (MoCA ≥ 23, ≤ 28, TMT B < 120 s) and long version (MoCA > 28, TMT B < 120 s). For each difficulty level, a NL puzzle was selected randomly from the pre-generated NL puzzles. The order of presentation of difficulty levels was randomized in order to avoid learning effects (van Kreveld et al., 2015 ). After completing the NL puzzle difficulty levels, participants evaluated the usability and their experience with the NL puzzles by filling in the SUS and the Perception of Game Training Questionnaire.

Statistical Analysis

For group comparison, only the NL difficulty levels from the short version, completed by all participants, were analyzed. Due to non-normally distributed data (visual inspection of histogram and quantile–quantile plots and Shapiro–Wilk tests), statistical differences between the participant groups were performed using the non-parametric Kruskal–Wallis test followed by multiple comparison post hoc tests using the pgirmess package (Giraudoux, 2018 ) in R version 1.1.463 (R Core Team, 2018 ). As criterion for statistical significance, a probability level of 0.05 was used in post hoc comparisons.

To assess the concurrent validity of the NL puzzle task, associations between NL puzzle performance measures and neuropsychological test measures of attentional, visuospatial and visuoconstructional, and executive function and global cognitive ability were tested by correlational analyses (Spearman rank correlation coefficients) using the sjstats package (Lüdecke, 2019 ). Furthermore, partial correlation analyses (using Spearman rank correlation coefficients), controlling for participant age, were performed using the ppcor package (Kim, 2015 ).

Difficulty level manipulation was examined using correlational analyses (Spearman rank correlation coefficients) between total time to solve the puzzle and difficulty level. Difficulty levels were ordered based on the set size (width × height) and the number of paths of the NL puzzles. Only data from the young and older adults who completed the full range of manipulated difficulty levels were analyzed using separate correlations for young and older adults.

Results for Demographic Variables and Neuropsychological Tests

Participant demographic variables and neuropsychological test data are shown in Table 1 . Groups differed significantly in age ( χ ( 4 ) 2 = 568.21; p < 0.001), but not in years of education ( F (4, 38) = 1.744, p = 0.161). Neuropsychological test measures revealed significant group differences in the TMT-A completion time ( χ ( 4 ) 2 = 29.43; p < 0.001) and errors ( χ ( 4 ) 2 = 11.26; p = 0.024), the TMT-B completion time ( χ ( 4 ) 2 = 32.13; p < 0.001) and errors ( χ ( 4 ) 2 = 13.04; p = 0.01) and number correct ( χ ( 4 ) 2 = 8.62; p = 0.07), the SMT solving time ( χ ( 4 ) 2 = 32.56; p < 0.001) and errors ( χ ( 4 ) 2 = 13.52; p < 0.01) and the MoCA score ( χ ( 4 ) 2 = 19.18; p < 0.001). Post hoc analyses (multiple comparison tests, p < 0.05) that young adults were significantly faster on the TMT-A and TMT-B than oldest adults (TMT-A p < 0.001, TMT-B p < 0.001) and patients with HD (TMT-A p = 0.021, TMT-B p = 0.019). Furthermore, young adults were significantly faster on the TMT-B than old adults ( p = 0.001). SMT solving time was significantly different between young adults and older adults ( p < 0.001), oldest adults and patients with HD ( p = 0.009).

Results for Perception and Usability of the NL Puzzle Game

Ratings of enjoyment ( χ ( 4 ) 2 = 3.81; p = 0.43), challenge ( χ ( 4 ) 2 = 8.75; p = 0.07), frustration ( χ ( 4 ) 2 = 8.98; p = 0.06), and motivation ( χ ( 4 ) 2 = 2.36; p = 0.67) for the NL puzzle game did not differ significantly between groups. There was a significant difference in usability ratings between groups ( χ ( 4 ) 2 = 12.04; p = 0.02), but post hoc comparisons failed to show any significant differences between groups (YA vs. OA, OOA vs. PD, OOA vs. HD, PD vs. HD: p = 1.000, YA vs. OOA p = 0.140, YA vs. PD p = 0.609, YA vs. HD p = 0.355, OA vs. OOA p = 0.228, OA vs. PD p = 0.689, OA vs. HD p = 0.414). Overall, individual SUS ratings ranged from 67.50 (“good”) to 100.00 (“best”) with a mean SUS rating of 86.43 (“excellent”; Bangor et al., 2008 ).

Results for NL Performance Measures

NL puzzle game performance measures by group can be found in Table 2 and are displayed in Figure 4 . For the time-based NL performance measures, there were significant group differences in TST ( χ ( 4 ) 2 = 419.51, p < 0.001), MET ( χ ( 4 ) 2 = 324.72, p < 0.001), and MPT ( χ ( 4 ) 2 = 410.97, p < 0.001). Post hoc analyses showed that both total solving (TST: YA vs. OA, YA vs. OOA, YA vs. PD, YA vs. HD, OA vs. OOA p < 0.001, OOA vs. PD p = 0.034, PD vs. HD: p = 0.002) and MPTs (MPT: YA vs. OA, YA vs. OOA, YA vs. HD, p < 1.000, YA vs. PD p = 0.009, OA vs. OOA p = 0.001, OOA vs. PD p = 0.015, OA vs. HD = 0.034, PD vs. HD: p = 0.008) were significantly different between all groups (multiple comparison test, p < 0.05), except between older adults and patients with PD (TST p = 1.000, MPT p = 1.000) and between oldest adults and patients with HD (TST p = 1.000, MPT p = 1.000). MET (MET: YA vs. OA, YA vs. OOA, YA vs. HD, p < 0.001, YA vs. PD p = 0.002, OA vs. OOA p = 0.003, OA vs. HD = 0.008, PD vs. HD: p = 0.043) differed significantly between all groups ( p < 0.05) with the exception of older adults and Parkinson’s patients, oldest adults and Parkinson’s patients, and oldest adults and Huntington’s patients that were not significantly different.

Time, accuracy, and movement-based performance measures for the Numberlink puzzle game by group.

All values, mean (SD), p value < 0.05 is significant. YA, Young Adults; OA, Older Adults; OOA, Old oldest Adults; PD, Parkinson’s Disease; HD, Huntington’s disease; TST, total solving time; MET, motor execution time; MPT, mental planning time; MVE, mean velocity; MDC, mean number of movement direction changes; Z comparison across all groups df = 4, A YA vs. OA, B YA vs. OOA, C YA vs. PD, D YA vs. HD, E OA vs. OOA, F OA vs. PD, G OA vs. HD, H OOA vs. PD, I OOA vs. HD, J PD vs. HD .

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Median solving times of Numberlink puzzles for young, older, and oldest adults, and patients with Parkinson’s and Huntington’s disease. Total solving time (TST) is shown separately as the time needed to connect paths motor execution time (MET) and time where no movements were made mental planning time (MPT), *significant difference at the 0.05 level.

Results for accuracy-based NL performance indicators further suggest significant group differences in the number of false ( χ ( 4 ) 2 = 50.03, p < 0.001) and excess moves ( χ ( 4 ) 2 = 42.44, p < 0.001) moves. There were no significant group differences in the number of used hints ( χ ( 4 ) 2 = 2.24, p = 0.69). The number of false moves were significantly different between the young and oldest adults ( p = 0.001), young adults and patients with HD ( p = 0.005), as well as between the older adults and the oldest adults ( p = 0.017) and the older adults and patients with HD. Post hoc comparisons further revealed that patients with HD made significantly more excess moves than both young ( p = 0.017) and older adults. Oldest adults further made significantly more excess moves than young adults ( p = 0.001).

Results for NL Puzzle Movement Metrics

Analysis of movement-based measures when playing the NL puzzles, revealed significant differences between groups in average movement velocity ( χ ( 4 ) 2 = 289.68, p < 0.001) and average number of MDC ( χ ( 4 ) 2 = 234.94, p < 0.001). Post hoc group comparisons revealed significant differences in average movement velocity (MVE: YA vs. OA, YA vs. OOA, YA vs. HD, p < 0.001, YA vs. PD p = 0.001, OA vs. OOA p = 0.029, OA vs. HD p = 0.008) between all groups ( p < 0.05) except with older adults and patients with PD, and oldest adults compared to patients with HD and PD. In terms of the number of MDC, all group comparisons (MDC: YA vs. OA p = 0.010, YA vs. OOA p < 0.001, YA vs. PD p = 0.028, YA vs. HD p = 0.001, OA vs. OOA p = 0.044, OA vs. HD: p = 0.013) were significant, except for patients with HD compared to oldest adults and patients with PD and patients with PD compared to older and oldest adults.

Correlation Between NL Puzzle Performance and Neuropsychological Test Measures

Correlational analyses (see Table 3 ) revealed significant associations between time-based NL puzzle performance (median TST) and performance in visuomotor and visuospatial (TMT-A time), executive (shifting, inhibition; TMT-B time), visuoconstructional and executive (planning and foresight) function (SMT time), and global cognitive ability (MoCA). Partial correlations controlling for age remained significant. Similarly, accuracy-based NL puzzle measures (number of false and excess moves) were significantly correlated with TMT-A and -B time, SMT time, and the MoCA score. However, when controlling for age with partial correlation, the correlations with SMT time and MoCA did not remain significant. Finally, movement-based measures from the NL puzzles (MVE, MDC) all showed significant correlations with SMT time, TMT-A and -B time, and MoCA score. After controlling for age, these correlations remained significant with the exception of correlations between MDC and global cognitive ability (MoCA).

Correlations between Numberlink puzzle performance and movement metrics and neuropsychological test measures.

SMT, Snellgrove maze task; TMT A, Trail making test part A; TMT B, Trail making test part B; MoCA, Montreal cognitive assessment; TST, total solving time; MVE, Mean velocity; MDC, mean number of movement direction changes; ns, not significant; ***significant at the 0.001 level, **significant at the 0.01 level, *significant at the 0.05 level .

Results for Task Difficulty Manipulation

Total NL puzzle solving time as a function of difficulty level are shown separately for young and older adults that completed the full range of NL difficulty levels in Figure 5 . Separate Spearman correlation analyses between total puzzle solving time and ranked difficulty levels revealed a significant positive association for both the young adults ( r (17) = 0.583, p < 0.001) and the older adults ( r (13) = 0.496, p < 0.001).

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Total Numberlink puzzle solving time as a function of ranked difficulty level for young ( n = 18) and older adults ( n = 14). Difficulty levels are defined by set size and the number of paths of the respective Numberlink puzzles.

The aim of this study was to evaluate the feasibility and preliminary validity of a maze-like NL puzzle video game as a tool to assess cognitive and motor differences in older adults and patients with NDDs. Results from this study show that NL puzzles are enjoyable, motivating, and user-friendly for older adults and patients with motor difficulties in neurodegenerative diseases. Analyses of game-based measures of cognitive and motor performance showed significant differences in executive and motor function within the different age groups and between the disease groups. However, there were no differences in total solving, mental planning, and MET between HD and the oldest age (OOA) group and between PD and the older age (OA) group. Initial concurrent validation showed that the NL puzzle game performance correlates with a standardized maze task (SMT) and tests for cognitive abilities relevant to cognitive decline due to aging and NDD. This study further shows that characteristics of the game can be adjusted to create graded levels of difficulty.

Analyses of performance on the played NL puzzle game levels showed significant age-group differences. Overall, older adults took longer to solve NL puzzles than young adults, and oldest adults were slower compared to older adults. The differences in NL puzzle performance between the age groups are consistent with age-related changes in motor and executive functioning (Zhang et al., 2007 ). One explanation for the effect of age on NL puzzle game performance might simply be a decline in motor processing speed with age (Salthouse, 2000 ; Ebaid et al., 2017 ). Findings from this study do show that MET (i.e., total time needed to manually connect paths in the maze) and average velocity of these movements in the NL puzzle game were significantly different between young, older, and oldest adults. However, the movement velocity and MDC did not differ between the oldest groups and the neurodegenerative disease groups. Reasons may lie in the visuoconstructional (Snellgrove, 2005 ) and perceptual-motor function (Zhao and Marquez, 2013 ) skills required for NL puzzle games. The perceptual-motor skills decline with age and neurodegenerative diseases. For PD and HD patients, the UPDRS-Motor score quantifies the decline in motor skills. It would be advisable to quantify the manual dexterity of older and oldest adults so that they can be adjusted for their game-based motor performances. PD patients recruited in our study were in their early disease stage and had very low UPDRS-Motor scores, which is seen clearly in their motor performances.

However, our finding of age-group differences in MPT (i.e., subtracting the MET from the TST) replicates previous studies that showed age effects on maze solving ability (Koss et al., 1991 ; Krishnasamy and Unsworth, 2011 ; de Souza et al., 2013 ), even when controlling for motor processing speed. The age effect on maze solving ability has been suggested to reflect age-related difficulties in executive and visuoconstructive functioning (Krishnasamy and Unsworth, 2011 ; de Souza et al., 2013 ). Interestingly, studies have suggested that executive deficits with aging are not general, but specifically affect planning ability that is usually measured using maze tasks (Zhang et al., 2007 ; Brandt et al., 2009 ) and strongly associated with everyday functioning in older age (Lewis and Miller, 2007 ). Furthermore, solving the NL puzzle game requires sequencing of motor actions to organize and sequence multiple paths that need to be connected in order to solve NL puzzles (Laurentiz, 2018 ). Similar to planning ability, sequencing motor actions is increasingly dependent on executive function and declines with age (Niermeyer et al., 2017 ).

Compared to young and older adults, patients with HD were significantly slower in both total solving, mental planning, and MET. This finding is underscored by the fact that HD patients made significantly more errors and excess moves when solving NL puzzles. Taken together, the difficulties of HD patients with solving maze-like NL puzzles are consistent with previous findings and reflect declines in psychomotor, visuospatial and executive function that occur throughout HD (Fedio et al., 1979 ; Montoya et al., 2006 ). Contrary to a previous study (Mimura et al., 2006 ), this study found no difference between PD patients and older adults in total solving, mental planning, and MET. This suggests that the PD patients in this study suffered no deficits in psychomotor and planning aspects of executive function. This finding was mirrored in the assessments done using the standardized SMT. PD patients recruited in our study had a good cognitive status, as indicated in their MoCA score (26.25 ± 3.40). Their MDS-UPDRS-III scores (17.75 ± 5.38) were significantly lower than the HD patients and displayed very few motor deficits. This was reflected in the game performance results such as mental planning and motor execution. However, PD medications can affect both their cognitive and motor skill levels, which cannot be ruled out.

As the MPT is TST minus the execution time, we could rule out a confounding effect of motor performance (Ebaid et al., 2017 ). In addition, we found identical group differences for game-based motor performance measures (MET and velocity and direction changes of movements), only this time, Parkinson’s patients were not different from oldest adults. Interestingly, these findings parallel earlier studies that assessed touch interactions on mobile devices in older adults and Parkinson’s patients (Keates and Trewin, 2005 ; Montague et al., 2014 ) and are supported by a recent viewpoint paper claiming that game-based movement measures reflect psychomotor deficits in age-related neurodegenerative diseases such as AD, PD, and HD (Mandryk and Birk, 2019 ).

The concurrent validity findings in this study show that game performance measures from the NL puzzle game were significantly correlated with performance on tests for visuomotor, visuospatial, executive, and visuoconstructional function. Game performance was also associated with global cognitive ability. These findings confirm recent suggestions that video games incorporate elements and cognitive challenges shared with psychological tasks (Holmgard et al., 2016 ). Furthermore, results indicate that game-based measures of cognitive and motor performance can be used to assess and monitor cognitive function in normal aging and neurodegenerative diseases (Koo and Vizer, 2019 ; Mandryk and Birk, 2019 ). Significant correlations between NL-based time and movement performance with tests of psychomotor, attentional, visuoconstructional, and executive functions support our assumption that NL puzzles share cognitive components with classical maze tasks (Snellgrove, 2005 ; Kirsch et al., 2006 ). Thus, a digital maze test like the NL can be used as a nonverbal intelligence test to estimate planning and inhibition. Moreover, they can overcome inconsistencies in interrater reliability seen in standard paper-based maze tests and can be deployed in a familiar user environment. The continuous and automatic data recording with digital technology will allow easy data collection of errors, corrections, and attempts of each user.

Our results on the usability and perception of the NL puzzles showed that they are enjoyable, motivating, user-friendly, and easy to use for both the healthy age groups and the neurodegenerative groups. However, there was a slight trend in rating the game session as challenging and frustrating, which is driven by perception ratings of the PD and HD patients. On the one hand, this might simply reflects the disease-related motor challenges in performance. On the other hand, it might be that it was harder for PD and HD patients to adapt to the difficulty levels. Adapting the difficulty levels to a person’s performance might resolve a part of this problem and also ensure that the puzzle games are equally challenging for players with different levels of cognitive ability (Lankoski, 2015 ; Holmgard et al., 2016 ).

Some limitations of this study need to be addressed: first, the sample size of this study is small, particularly for the groups with neurodegenerative disorders. Moreover, the level of progression of disease within the different neurodegenerative disease was not controlled for in this study. Therefore, validation studies with larger sample size are definitely needed to further confirm these initial findings. Second, our findings are based on comparing the performance on a limited set of NL puzzle difficulty levels. A possible solution to this would be to use computerized adaptive cognitive testing that adapts the difficulty of the task to the individual’s level of cognitive ability by selecting test items from graded difficulty levels. Using the pre-generated NL puzzle difficulty levels from this study, this would allow to better detect subtle changes in executive and motor function across a wider range of maze-like NL puzzle difficulties and avoid both floor and ceiling effects. Third, this study is cross-sectional, and participants play the NL only once and for the first time.

To sum up, this study supports recent suggestions that game-based data from playing a commercial maze-like puzzle game provides potential “digital biomarkers” to assess cognitive function and cognitive decline over time. In particular, NL puzzle games seem promising for capturing changes in visuomotor, visuoconstructional, and executive function related to aging and neurodegeneration.

Data Availability Statement

Ethics statement.

The studies involving human participants were reviewed and approved by The ethics committee of Canton Bern and Ethics Committee of Northwest and Central Switzerland. The patients/participants provided their written informed consent to participate in this study.

Author Contributions

TN, RM, and PU contributed to the conception and organization of the research. AC, TN, TV, JM, J-MB, and PU participated in the execution and data collection. AC, NS, AB, and PU designed the data analysis and statistical methods. AC wrote the first draft of the manuscript. All authors participated in the review and critical review of the manuscript.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Acknowledgments

We would like to thank the residents and staff at the Senevita Bornblick senior residence home in Olten, Switzerland and the Burgerspittel, Bern, Switzerland. The authors extend their gratitude to Luca Persello for data handling and to Julia Truttmann and Manuela Wüthrich for their valuable help with recruiting and testing participants.

Funding. The study was partially supported by a Novartis–Freenovation grant. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Supplementary Material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fnagi.2020.00087/full#supplementary-material .

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Data Structures
Linked List
Binary Tree
Binary Search Tree
Segment Tree
Disjoint Set Union
Fenwick Tree
Red-Black Tree
Advanced Data Structures
Backtracking Algorithm
Introduction to Backtracking - Data Structure and Algorithm Tutorials
Difference between Backtracking and Branch-N-Bound technique
What is the difference between Backtracking and Recursion?

Standard problems on backtracking

The Knight's tour problem

Rat in a Maze

N Queen Problem
Subset Sum Problem using Backtracking
M-Coloring Problem
Hamiltonian Cycle
Algorithm to Solve Sudoku | Sudoku Solver
Magnet Puzzle
Remove Invalid Parentheses
A backtracking approach to generate n bit Gray Codes
Permutations of given String

Easy Problems on Backtracking

Print all subsets of a given Set or Array
Check if a given string is sum-string
Count all possible Paths between two Vertices
Find all distinct subsets of a given set using BitMasking Approach
Find if there is a path of more than k length from a source
Print all paths from a given source to a destination
Print all possible strings that can be made by placing spaces

Medium prblems on Backtracking

8 queen problem
Combinational Sum
Warnsdorff's algorithm for Knight’s tour problem
Find paths from corner cell to middle cell in maze
Find Maximum number possible by doing at-most K swaps
Rat in a Maze with multiple steps or jump allowed
N Queen in O(n) space

Hard problems on Backtracking

Power Set in Lexicographic order
Word Break Problem using Backtracking
Partition of a set into K subsets with equal sum
Longest Possible Route in a Matrix with Hurdles
Find shortest safe route in a path with landmines
Printing all solutions in N-Queen Problem
Print all longest common sub-sequences in lexicographical order
Top 20 Backtracking Algorithm Interview Questions

We have discussed Backtracking and Knight’s tour problem in Set 1 . Let us discuss Rat in a Maze as another example problem that can be solved using Backtracking.

Consider a rat placed at (0, 0) in a square matrix of order N * N . It has to reach the destination at (N – 1, N – 1) . Find all possible paths that the rat can take to reach from source to destination. The directions in which the rat can move are ‘U'(up) , ‘D'(down) , ‘L’ (left) , ‘R’ (right) . Value 0 at a cell in the matrix represents that it is blocked and rat cannot move to it while value 1 at a cell in the matrix represents that rat can be travel through it. Return the list of paths in lexicographically increasing order. Note : In a path, no cell can be visited more than one time. If the source cell is 0 , the rat cannot move to any other cell.

Input: Output: DRDDRR Explanation:

Rat in a Maze using Backtracking :

We use a backtracking algorithm to explore all possible paths . While exploring the paths we keep track of the directions we have moved so far and when we reach to the bottom right cell , we record the path in a vector of strings.

Step-by-step approach:

Create isValid() function to check if a cell at position (row, col) is inside the maze and unblocked.
Base case: If the current position is the bottom-right cell, add the current path to the result and return.
Mark the current cell as blocked.
Calculate the next position based on the current direction.
If the next position is valid (i.e, if isValid() return true ), append the direction to the current path and recursively call the findPath() function for the next cell.
Backtrack by removing the last direction from the current path.
Mark the current cell as unblocked before returning.

Below is the implementation of the above approach:

Time Complexity: O(3^(m*n)), because on every cell we have to try 3 different directions. Auxiliary Space: O(m*n), Maximum Depth of the recursion tree(auxiliary space).

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Nvidia and Microsoft are solving a big problem with Copilot+

The Surface Laptop running local AI models.

When Microsoft announced Copilot+ PCs a few weeks back, one question reigned supreme : Why can’t I just run these AI applications on my GPU? At Computex 2024 , Nvidia finally provided an answer.

Nvidia and Microsoft are working together on an Application Programming Interface (API) that will allow developers to run their AI-accelerated apps on RTX graphics cards . This includes the various Small Language Models (SLMs) that are part of the Copilot runtime, which are used as the basis for features like Recall and Live Captions.

With the toolkit, developers can allow apps to run locally on your GPU instead of the NPU. This opens up the door to not only more powerful AI applications, as the AI capabilities of GPUs are generally higher than NPUs, but also the ability to run on PCs that don’t currently fall under the Copilot+ umbrella.

It’s a great move. Copilot+ PCs currently require a Neural Processing Unit (NPU) that’s capable of at least 40 Tera Operations Per Second (TOPS). At the moment, only the Snapdragon X Elite satisfies that criteria. Despite that, GPUs have much higher AI processing capabilities, with even low-end models reaching to 100 TOPS, and higher-end options scaling even higher.

In addition to running on the GPU, the new API adds retrieval-augmented generation (RAG) capabilities to the Copilot runtime. RAG gives the AI model access to specific information locally, allowing it to provide more helpful solutions. We saw RAG on full display with Nvidia’s Chat with RTX earlier this year.

Outside of the API, Nvidia announced the RTX AI Toolkit at Computex. This developer suite, arriving in June, combines various tools and SDKs that allow developers to tune AI models for specific applications. Nvidia says that by using the RTX AI Toolkit, developers can make models four times faster and three times smaller compared to using open-source solutions.

We’re seeing a wave of tools that enable developers to build specific AI applications for end users. Some of that is already showing up in Copilot+ PCs, but I suspect we’ll see far more AI applications at this point next year. We have the hardware to run these apps, after all; now we just need the software.

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It's not just you -- Microsoft is putting Copilot just about anywhere, and now it's available on Telegram as a bot that's part of its "copilot-for-social" project. Windows Latest reports that to use the bot, you'll need to confirm your phone number by sending your contact in the chat. The company is expanding its AI integration into one of the most popular messaging apps, but it might not end there.

Microsoft claims that it won't save your phone number for anything other than verification, but it's needed because the service is currently unavailable for anyone in the EU, at least for now. If anyone in the EU tries to access it, they will be blocked.

We were just inundated with a whole slew of new Windows laptops. These aren't your average notebooks, though -- they're part of Microsoft's new Copilot+ PC program -- built from the ground up with AI and ARM chips in mind.

A lot of these laptops emphasize the same performance and battery life gains caused by Qualcomm's Snapdragon X chips. Those are impressive, but comparing all these laptops against each other, it's Samsung's entry that makes me the most excited. Design efficiency

Imagine a laptop with an iteration of Nvidia’s ARM-based CPU combined with a powerful RTX graphics card, all enhanced by AI. Years ago, that would have sounded outlandish, but now it seems like it could actually happen.

In a recent interview with Bloomberg, Nvidia CEO Jensen Huang and Dell CEO Michael Dell more or less confirmed that Team Green will enter the AI-PC hype next year.

IMAGES

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Mazes For Kids Ages 8-12: Maze Activity Book 8-10, 9-12, 10-12 year
FREE Maze Solving Equations Activities ⋆ Algebra 1 Coach
Problem Solving with Mazes and Crayon
24 DIY Problem Solving Mazes For Kids (Craft Gossip)
Completing mazes is a great way for children to enhance their problem

VIDEO

A Maze a Day
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🔍Find The Answer (Maze Challenge) -🕹Marble Maze Puzzle Game
Robot Maze Solver
The Art of Puzzle solving #maze #puzzle #viral #animation
#maze # 84

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Rat in a Maze
Let us discuss Rat in a Maze as another example problem that can be solved using Backtracking. Consider a rat placed at (0, 0) in a square matrix of order N * N. It has to reach the destination at (N - 1, N - 1). Find all possible paths that the rat can take to reach from source to destination. The directions in which the rat can move are ...
Nvidia and Microsoft are solving a big problem with Copilot+
Best Video Games. Best Mac Games; Best PS5 Games; Best Games on PS Plus; ... Nvidia and Microsoft are solving a big problem with Copilot+ By Jacob Roach June 2, 2024 5:30AM Luke Larsen / Digital ...