I haven't seen exactly that here in the US, but I've got some puzzles that aren't just knobs on 4 sides. Quite a few of the pieces are slanted, have 3 sides, sit at an angle, etc. It adds quite a lot of variation without breaking the interlocking nature of the puzzles we are used to.
Edit: I just realized that for kids, those knobs are easy to break off. The puzzle you showed makes more sense for kids.
My family has a collection of home-made jigsaw puzzles (literally made by a guy cutting up a wood-backed picture with a jigsaw) which look similar in terms of shape to that (but more pieces, 500 or so). I think they are much more interesting to solve, especially since the contours often follow the shapes in the image, which adds some twists in the difficulty. I would also be interested in finding a way to buy such puzzles nowadays (especially since they would in priciple be easy to produce through laser-cutting).
Don't know if this is originally japanese, but we have some puzzles like that in germany. One we have at home has a frame background image different to the puzzle motive. Adding visual disturbance. My kids seemed not to be disturbed as much by it than their parents in the first try
The grid for jigsaw pieces is normally not square, at least when I used to solve them. Something like 3:2 was more typical.
If you can vary the aspect ratio of the average piece between 2:3 and 3:2, or even just pick one of the extremes so you can rotate the grid, a lot more numbers are possible. 1000 can be an almost-square grid.
None of this has anything to do with the mathematical content which is really the point of the blog post, but it's where my mind went when he claimed certain numbers make for bad jigsaw sizes.
It seems too long and skinny to me too! But I needed a nice place to draw the line. And I was originally doing the calculations in my head, so allowing up to only 1.5:1 or something would have made the arithmetic a bit harder.
The middle part is a bit too heady for me but I'm fascinated by this sentence in the conclusion: "I assume that you wouldn’t actually see a 999-piece puzzle because the lawyers would claim calling that 1000 was false advertising."
My soul wants to hope that nobody would be as crazy as to sue a puzzle company for having 999 pieces instead of an even 1000 but my mind is saying "duh, of course they would".
Wonder what the situation is like for those new types of puzzles that are circular or have irregular shapes. Those are probably easier to make into an even, pretty number but I feel like there would still be exceptions.
Call me strange but I've actually counted the pieces to a few of my kids puzzles and the 100 and 200 piece puzzles never have exactly that number. And it wasn't off-by-one either. I always figured it make sense that the box advertises an 'around'-number as an indication of difficulty. (For the Why?: When puzzling together the key is to not overdo with the helping, so there is some time at hand for other brain activities.)
I wasted a lot of time as a kid looking for missing pieces (and recounting them several times to begin with) in a 2000 puzzle that didn't have that many pieces. I wish they had an exact number in the fine print.
It implies that more puzzle pieces is better, so somebody might sue if they were to get fewer pieces than indicated on the box.
But how about suing because you get more pieces than advertised? Having more pieces than what's written on the box makes it harder to do the puzzle, causing stress, anxiety and lost time!
Take them greedy puzzle companies to court, left and right!
A related question I once thought of: What’s the minimum number of pieces you need to assemble to form a “skeleton” of a puzzle - such that every other piece fits directly to one of the skeleton pieces?
For an 9x4 puzzle I came up with the following pattern - a skeleton that uses 15 of 36 pieces:
-X—-X—-X-
-X—-X—-X-
XXXXXXXXX
—————————
It would be interesting to see if that’s optimal, and if the optimal solution is different for different puzzle sizes.
An (obvious?) iterative improvement on this solution is to just take the first two rows and mirror it for the bottom two, for a solution of 4x3=12 pieces.
-X--X--X-
-X--X--X-
-X--X--X-
-X--X--X-
The naive lower bound on a solution is N/5, based on the maximum covering a piece might have (of course, that's not tight, due to literal corner cases). Heuristically, we might expect an optimal solution to look like E[1/K]≈8.6, where K is defined as the number of neighbors a square has.
Attempting to manually reduce the overlaps, I came up with
-X--X--X-
---X-X---
X-------X
--X-X-X--
for 10.
(Of course, the puzzle may be underspecified, since if you also require that your skeleton forms a connected component, then neither of these solutions count.)
edit: And, one more manual tiling (10 again) which exhibits better periodicity:
That's an interesting answer! Your final layout looked oddly familiar to me, and it finally occurred to me: The problem of finding a skeleton puzzle was very similar to finding an optimal layout for a sugarcane farm in Minecraft, that my high school friends and I experimented with years ago. Each square of sugarcane needs to be connected to at least one square of water, and the most efficient farm is the one requiring the least amount of space for water.
https://i.imgur.com/T5yh9.jpg
That's actually an NP-complete problem, called the Dominating Set problem[0]. One application of this problem is used in OLSR[1], a routing protocol used in Mobile Ad-Hoc Networks (MANET). To see how it is useful for MANETs, you can look at the problem in a different, but equivalent, way. Namely, try to find the smallest subset D of nodes (puzzle pieces), such that every two nodes that are not in D are connected to each other through a node in D. In the case of MANETs, these nodes are wireless stations. The nodes in the Domating Set are chosen to relay broadcasting messages for their neighbours. This way congestion on the medium is reduced but every node can still send and receive broadcasts.
It turns out that the answer to this question is known for all sizes of grid[1], though it was found fairly recently (2011) and the answer for small grids is quite complicated.
For the 9x4 grid, the optimal number is indeed 10.
For large n×m grids, with n and m ≥ 16, the minimum number is:
floor((n + 2)(m + 2) / 5) - 4.
The linked paper has a full-page table giving the formulae for small grids.
Thanks to Schoolmeister for mentioning the name of this problem, which gave me the right terms to search for.
I love that site. I'm currently working on the earth one. The spherical nature of the content means you have to wrangle with map projection stuff--which I thought I understood, but dealing with it in a puzzling setting is a whole new animal.
Of course the most interesting jigsaw puzzles don’t follow this constraint. Take for example Yuu Asaka’s amazing _Jigsaw puzzle 19_, which consists of 19 pieces – and they are all corner pieces!
Seems weird that they wouldn't put the exact number on the box. Feel like that'd seem more interesting than a bunch of zeros. Are people more likely to buy a box saying 1000 than 1008?
I think because it's not about the actual number of pieces. The piece ranks being rounded turns them into an approximate difficulty rating that is easily understood. It also (roughly) shows the size of the puzzle and the amount of time it would take to complete.
Just realized that in a 2d puzzle you have only 4 corner pieces, but in a higher-dimensional (say, 20d) puzzle, almost all pieces would be corner pieces.
I agree, don't think it works out. Consider a 3x3...x3 puzzle (with N dimensions). There are 2^N corners but 3^N pieces so the corner-to-piece ratio decreases exponentially in N.
However, what does "corner" mean in an N dimensional puzzle? I was taking it to mean that it has N sides without pegs or holes. But if you took it to mean at least 2 sides without pegs or wholes, then indeed the ratio would increase (I think there are 3^N - 2^N - 1 of those) with N. But I think this is the less-satisfactory way to generalize "corner" pieces and these really should be thought of as edge pieces.
Ah, this is a nice point, it does work for the number of edge pieces. In this case (3x3x...x3) every piece but one is an edge piece!
More generally, with k pieces on a side, there are k^N pieces total, and (k-2)^N internal pieces, so in the limit, the ratio of internal to total is (1-2/k)^N -> e^{-2N/k}.
So if the number of dimensions gets to be bigger the grid size, almost all pieces are "edge" (i.e. outer) pieces. Even if N=k/2, something like 70% of the pieces are edge pieces.
In a 2d, rectangular puzzle, with boring die-cut pieces, you have only 4 corner pieces. :-)
Puzzles at higher difficulties are not rectangular, don't have straight sides, or have straight & corner-looking pieces in the middle.
My favorite difficult puzzles are Stumpcraft -- this small business makes wooden, beautiful smelling puzzles with a laser-cutting technique. They're very difficult with intricate and intentionally misleading pieces and geometries. https://www.stumpcraft.com/
Ha. That sounded close to me but it's not right, I think the reason is that the corners are very small.
Suppose a perfect grid in a hypercube. In d dimensions we have 2^d corner pieces. But if there are k pieces per side, then k^d total pieces (for example 10^d).
If we consider pieces adjacent to a corner, including a diagonal, then these are little cubes of size 2^d sitting at the corners, so a total of (2^d)(2^d) = 4^d pieces close to a corner.
In general taking little cubes out of the corners isn't enough to get most of the pieces. If we think about the diagonals of length k sqrt(d), we've only cut those a little shorter.
We did this moon puzzle [0] towards the start of quarantine and it was _super_ hard. Part of it was the same-ness of all the featureless grey. It was slow going.
I’d guess because you could easily build a machine to cut the pieces with long strips of metal rather than welding together a bunch of disjointed pieces.
They don't always, but the obvious/boring thing to do is to start with a shape that tiles the plane, like rectangles or hexagons. A Penrose tiling might be an interesting approach to take.
I discovered these a few years back: https://n-e-r-v-o-u-s.com/shop/ and I really dig 'em. They might be using a grid in some cases--but if so it's pretty hard to tell.
I'm working on my fourth. It's hard to worry about the apocalypse when the complexity of the puzzle requires your whole brain to make progress.
I had thought I had read somewhere about the challenges of designing a unique and valid puzzle (so that pieces couldn't fit together that shouldn't fit together) with the note that puzzle manufacturers apparently made new puzzle piece grids for each puzzle. I was pretty sure it was on kottke, and instead I found the exact opposite of that, with a link to the following project of mixing pieces together from puzzles to create surreal artwork:
https://puzzlemontage.crevado.com
I was hoping this would be about how difficulty scales with piece count (I doubt it's linear), and what a practical limit for piece count would be in terms of the time to solve it. If I had to guess, it would be n^2.
Anecdotally, my grandma finds 500 piece puzzles a lot harder than 300.
Hmm, would it have to do with information density? If you had a puzzle that was a simple gradient, you could easily locate the region a piece belonged to despite the number of pieces. If the puzzle was blank, the difficulty would ramp up much quicker.
n^2 seems reasonable to me as well. If you have n pieces, you have to make O(n) connections to solve the puzzle. But to find the piece that makes each connection, you need to look through O(n) pieces. Multiplying gives O(n^2).
> My 7 yo pointed out that our 300 piece puzzle actually contains 18 x 18 = 324 pieces and I just don’t know what to believe anymore.
Did they actually count the pieces? In my experience, puzzles are rarely/never simply exact columns and rows of pieces such that you can count the outside edge pieces and multiply to get the number of pieces. Instead there are pieces of all sorts of shapes that create interesting patterns within the puzzle which could result in a number much smaller than multiplying the outside pieces might indicate.
Some items get re-upped, and the timestamps get "rolled back" in interesting ways. Eventually it all comes out in the wash, but it can lead to local (space and time) apparent inconsistencies like this.
Don't sweat the small stuff... and it's all small stuff. -- Richard Carlson
This is just a random puzzle and not something I hand picked to make a point. As you can see the pieces are no where near a rigid row/column layout. In my experience this is the norm.
67 comments
[ 3.8 ms ] story [ 143 ms ] threadhttps://m.youtube.com/watch?v=f5XdPqMJVQM
I don't know if this is a specific Japanese style, but I never saw puzzle like this "in the west".
This one has a frame, but they also make some without (pieces that interlock like our puzzle), still with those weird shapes.
I really love them:)
You could probably generate the shapes automatically via something like voronoi diagram cells plus some rounding of corners?
Edit: I just realized that for kids, those knobs are easy to break off. The puzzle you showed makes more sense for kids.
If you want to go overboard, you can probably use some machine learning to decide on the curves for an arbitrary input image.
If you can vary the aspect ratio of the average piece between 2:3 and 3:2, or even just pick one of the extremes so you can rotate the grid, a lot more numbers are possible. 1000 can be an almost-square grid.
None of this has anything to do with the mathematical content which is really the point of the blog post, but it's where my mind went when he claimed certain numbers make for bad jigsaw sizes.
It seems too long and skinny to me too! But I needed a nice place to draw the line. And I was originally doing the calculations in my head, so allowing up to only 1.5:1 or something would have made the arithmetic a bit harder.
My soul wants to hope that nobody would be as crazy as to sue a puzzle company for having 999 pieces instead of an even 1000 but my mind is saying "duh, of course they would".
Wonder what the situation is like for those new types of puzzles that are circular or have irregular shapes. Those are probably easier to make into an even, pretty number but I feel like there would still be exceptions.
But how about suing because you get more pieces than advertised? Having more pieces than what's written on the box makes it harder to do the puzzle, causing stress, anxiety and lost time!
Take them greedy puzzle companies to court, left and right!
For an 9x4 puzzle I came up with the following pattern - a skeleton that uses 15 of 36 pieces:
It would be interesting to see if that’s optimal, and if the optimal solution is different for different puzzle sizes.Attempting to manually reduce the overlaps, I came up with
for 10.(Of course, the puzzle may be underspecified, since if you also require that your skeleton forms a connected component, then neither of these solutions count.)
edit: And, one more manual tiling (10 again) which exhibits better periodicity:
https://gist.github.com/cipherboy/dc14769830d74a0f783c7ae29b...
[0] https://en.wikipedia.org/wiki/Dominating_set
[1] https://en.wikipedia.org/wiki/Optimized_Link_State_Routing_P...
For the 9x4 grid, the optimal number is indeed 10.
For large n×m grids, with n and m ≥ 16, the minimum number is:
floor((n + 2)(m + 2) / 5) - 4.
The linked paper has a full-page table giving the formulae for small grids.
Thanks to Schoolmeister for mentioning the name of this problem, which gave me the right terms to search for.
1. https://arxiv.org/abs/1102.5206
https://n-e-r-v-o-u-s.com/shop/product.php?code=346
Edit: To clarify it relates to the title, rather than the content of the article. :D
Highly recommended.
https://www.puzzlemaster.ca/browse/novelty/packing/12838-jig...
btw: the are 4 similar jigsaw solved by mr. puzzle, you can find here[1], here[2] and here[3]
[0] https://www.youtube.com/watch?v=BPearqSivSc
[1] https://www.youtube.com/watch?v=H7xJePIvYbA
[2] https://www.youtube.com/watch?v=2-clIQ0NS6o
[3] https://www.youtube.com/watch?v=-9Hl0zMMwLw
However, what does "corner" mean in an N dimensional puzzle? I was taking it to mean that it has N sides without pegs or holes. But if you took it to mean at least 2 sides without pegs or wholes, then indeed the ratio would increase (I think there are 3^N - 2^N - 1 of those) with N. But I think this is the less-satisfactory way to generalize "corner" pieces and these really should be thought of as edge pieces.
More generally, with k pieces on a side, there are k^N pieces total, and (k-2)^N internal pieces, so in the limit, the ratio of internal to total is (1-2/k)^N -> e^{-2N/k}.
So if the number of dimensions gets to be bigger the grid size, almost all pieces are "edge" (i.e. outer) pieces. Even if N=k/2, something like 70% of the pieces are edge pieces.
Puzzles at higher difficulties are not rectangular, don't have straight sides, or have straight & corner-looking pieces in the middle.
My favorite difficult puzzles are Stumpcraft -- this small business makes wooden, beautiful smelling puzzles with a laser-cutting technique. They're very difficult with intricate and intentionally misleading pieces and geometries. https://www.stumpcraft.com/
Suppose a perfect grid in a hypercube. In d dimensions we have 2^d corner pieces. But if there are k pieces per side, then k^d total pieces (for example 10^d).
If we consider pieces adjacent to a corner, including a diagonal, then these are little cubes of size 2^d sitting at the corners, so a total of (2^d)(2^d) = 4^d pieces close to a corner.
In general taking little cubes out of the corners isn't enough to get most of the pieces. If we think about the diagonals of length k sqrt(d), we've only cut those a little shorter.
I highly recommend the Mars puzzle in particular[1].
They also _smell_ wonderful. Like fresh cut wood.
[0] https://bewilderness-puzzles.com/
[1] https://bewilderness-puzzles.com/collections/circular-puzzle...
[0]: https://fourpointpuzzles.com/products/the-moon
Possibly the work of Jason Jigs:
https://www.youtube.com/watch?v=NV0zn8YVTAM
Here's a short time-lapse of the building process
[0]: https://twitter.com/shimmmaz/status/1299393187304280066?s=20
Of course, this doesn't answer "Why 90 degrees?"
I discovered these a few years back: https://n-e-r-v-o-u-s.com/shop/ and I really dig 'em. They might be using a grid in some cases--but if so it's pretty hard to tell.
I'm working on my fourth. It's hard to worry about the apocalypse when the complexity of the puzzle requires your whole brain to make progress.
Anecdotally, my grandma finds 500 piece puzzles a lot harder than 300.
n^2 seems reasonable to me as well. If you have n pieces, you have to make O(n) connections to solve the puzzle. But to find the piece that makes each connection, you need to look through O(n) pieces. Multiplying gives O(n^2).
Did they actually count the pieces? In my experience, puzzles are rarely/never simply exact columns and rows of pieces such that you can count the outside edge pieces and multiply to get the number of pieces. Instead there are pieces of all sorts of shapes that create interesting patterns within the puzzle which could result in a number much smaller than multiplying the outside pieces might indicate.
In my experience, they exactly are.
Of course, there are plenty of "unconventional" ones too, but the majority has rigid row/column layout.
Some items get re-upped, and the timestamps get "rolled back" in interesting ways. Eventually it all comes out in the wash, but it can lead to local (space and time) apparent inconsistencies like this.
Don't sweat the small stuff... and it's all small stuff. -- Richard Carlson
https://imgur.com/rUrTPC3
This is just a random puzzle and not something I hand picked to make a point. As you can see the pieces are no where near a rigid row/column layout. In my experience this is the norm.
But we have hundreds of puzzles from tons of manufacturers. Other than the puzzles for little kids, all of our puzzles are non-grids.
Go try a pre-1950s 'pasttime puzzle' (wood). Most are rectangular, but certainly not some absurd same-length rows and columns.