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How many pieces can a puzzle have?
- eru 6y agoThe number theory aspects here are interesting. In practice, a puzzle that slightly cheats the grid might also be interesting.
- kuon 6y agoMy wife is Japanese and our kids had a lot of puzzle with very weird shapes, like this. https://m.youtube.com/watch?v=f5XdPqMJVQM https://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:)
- eru 6y agoInteresting! Yes, something like that would be relatively easy to make for any number of pieces. You could probably generate the shapes automatically via something like voronoi diagram cells plus some rounding of corners?
- wccrawford 6y agoI 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.
- rcxdude 6y agoMy 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).
- eru 6y agoMight be a nice business idea! If you want to go overboard, you can probably use some machine learning to decide on the curves for an arbitrary input image.
- seedie 6y agoDon'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
- dmurray 6y agoThe 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.
- beervirus 6y agoHe was allowing for aspect ratios all the way out to 2:1 (which seems too long and skinny to me).
- messe 6y agoThat's for the entire puzzle. I think they meant that the pieces themselves have closer to 3:2 aspect ratios.
- beervirus 6y agoOops, I misread.
- madcaptenor 6y agoOP here. 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.
- ferros 6y agoPuzzles are a great way to pass that Corona time at home.
- ConsiderCrying 6y agoThe 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.
- mvn9 6y agoJust cut one piece in half. It's a puzzle.
- wjnc 6y agoCall 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.)
- foobarian 6y agoI 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.
- skipnup 6y agoThe last puzzle I solved actually had a small table with the advertised and actual number of pieces.
- w-m 6y agoIt 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!
- staz 6y agojust realised if you want "squarish" piece you can't really have a rectangular puzzle with a prime number of pieces without resorting to tricks
- pansa2 6y agoA 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.
- vitus 6y agoAn (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: --X---X-- X---X---X --X---X-- X---X---X
- cipherboy 6y agoI don't think nine is possible, only 10. https://gist.github.com/cipherboy/dc14769830d74a0f783c7ae29b8682e0 https://gist.github.com/cipherboy/dc14769830d74a0f783c7ae29b...
- mahathu 6y agoThat'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 https://i.imgur.com/T5yh9.jpg
- haar 6y agoSomewhat related, but I thoroughly enjoy my infinite galaxy jigsaw puzzle: https://n-e-r-v-o-u-s.com/shop/product.php?code=346 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
- __MatrixMan__ 6y agoI 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. Highly recommended.
- lepton 6y agoAh, but how many sequences of 3 evenly spaced numbers does the sequence have?
- robinhouston 6y agoOf 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! https://www.puzzlemaster.ca/browse/novelty/packing/12838-jigsaw-puzzle-19 https://www.puzzlemaster.ca/browse/novelty/packing/12838-jig...
- jacquesm 6y agoNice but super simple, the tips are unique which makes it easy to solve. Still, very original.
- _ZeD_ 6y agonot so simple. If you want to "spoiler" the solution, you can find the mr. puzzle video on youtube[0]. 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 https://www.youtube.com/watch?v=BPearqSivSc [1] https://www.youtube.com/watch?v=H7xJePIvYbA https://www.youtube.com/watch?v=H7xJePIvYbA [2] https://www.youtube.com/watch?v=2-clIQ0NS6o https://www.youtube.com/watch?v=2-clIQ0NS6o [3] https://www.youtube.com/watch?v=-9Hl0zMMwLw https://www.youtube.com/watch?v=-9Hl0zMMwLw
- __s 6y agoSeems 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?
- Tokkemon 6y agoI 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.
- david_draco 6y agoJust 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.
- pfortuny 6y agoWhy? Are you not missing the “interior”?
- tgb 6y agoI 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.
- bo1024 6y agoAh, 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.
- mfenniak 6y agoIn 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/ https://www.stumpcraft.com/
- cdubzzz 6y agoSomewhat related — we recently purchased and have enjoyed a couple of circular puzzles from a small puzzle maker in the PNW — Bewilderness Puzzles[0]. I highly recommend the Mars puzzle in particular[1]. They also _smell_ wonderful. Like fresh cut wood. [0] https://bewilderness-puzzles.com/ https://bewilderness-puzzles.com/ [1] https://bewilderness-puzzles.com/collections/circular-puzzles/products/mars-wooden-puzzle-266-pieces-round-jigsaw-with-crazy-alien-cut-pattern-for-grown-up-puzzle-lovers-original-artwork-by-erin-darling https://bewilderness-puzzles.com/collections/circular-puzzle...
- shimmmaz 6y agoA few days ago we got the "Milky-Way at night" circular puzzle from bewilderness. It was quite the experience. Here's a short time-lapse of the building process [0]: https://twitter.com/shimmmaz/status/1299393187304280066?s=20 https://twitter.com/shimmmaz/status/1299393187304280066?s=20
- caseysoftware 6y agoWow. Some of those are beautiful. Thanks for sharing. I have a 5yo who will owe you a thank you too. :)
- Xavdidtheshadow 6y agoWe 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. [0]: https://fourpointpuzzles.com/products/the-moon https://fourpointpuzzles.com/products/the-moon
- dane-pgp 6y ago> Part of it was the same-ness of all the featureless grey. Possibly the work of Jason Jigs: https://www.youtube.com/watch?v=NV0zn8YVTAM https://www.youtube.com/watch?v=NV0zn8YVTAM
- cdubzzz 6y agoWow, 1,000 pieces for that is intense. The Mars one we got was tough for similar reasons, but didn't take too long since it is so much smaller.
- noja 6y agoStupid question: why does a puzzle follow a grid?
- justwalt 6y agoI’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.
- mqus 6y agoI would assume that grids are producing pieces that are most similar to another, making the puzzle more difficult, while also being easy to design. Of course, this doesn't answer "Why 90 degrees?"
- __MatrixMan__ 6y agoThey 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/ 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.
- dhosek 6y agoI 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 https://puzzlemontage.crevado.com
- dhosek 6y agoI think this is the article that was referenced by the blog post I read. It implies that new cutters are made for each puzzle (which is contradicted by the above). https://www.nytimes.com/2020/04/08/business/coronavirus-jigsaw-puzzles.html https://www.nytimes.com/2020/04/08/business/coronavirus-jigs...
- dehrmann 6y agoI 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.
- jayd16 6y agoHmm, 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.
- madcaptenor 6y agoI wrote the original post. 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).
- irrational 6y ago> 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.
- fireattack 6y ago> In my experience, puzzles are rarely/never simply exact columns and rows In my experience, they exactly are. Of course, there are plenty of "unconventional" ones too, but the majority has rigid row/column layout.
- fireattack 6y agoUnrelated but @dang: I made this comment 1 hour ago and it shows so in https://news.ycombinator.com/reply?id=24212997 https://news.ycombinator.com/reply?id=24212997. However when I check the thread https://news.ycombinator.com/item?id=24197668 https://news.ycombinator.com/item?id=24197668 it shows "10 min ago" (the one I'm replying to has the same issue.) Is this just cache or a bug?
- ColinWright 6y agoNot @dang, not even a mod, but ... 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
- fireattack 6y agoThanks for the reply. Yeah not a big deal at all, just curious.
- mauvehaus 6y agoYou see this in product packaging too: Domino sugar cubes come 198 to a box. This yields a much more pleasing and practical box than 200.
- frandroid 6y agoIs the box labelled "Almost 200 pieces"?
- rcgorton 6y agoThe author incorrectly assumes: - completely rectangular - all columns/rows have the same 'length' Go try a pre-1950s 'pasttime puzzle' (wood). Most are rectangular, but certainly not some absurd same-length rows and columns.