How many pieces can a puzzle have?
gottwurfelt.com
gottwurfelt.com
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
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...
Possibly the work of Jason Jigs:
Here's a short time-lapse of the building process
[0]: https://twitter.com/shimmmaz/status/1299393187304280066?s=20
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
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.
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.
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
—————————
-X--X--X-
-X--X--X-
-X--X--X-
-X--X--X-
Attempting to manually reduce the overlaps, I came up with
-X--X--X-
---X-X---
X-------X
--X-X-X--
(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---Xhttps://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.
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!
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.
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).
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:)
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.
Go try a pre-1950s 'pasttime puzzle' (wood). Most are rectangular, but certainly not some absurd same-length rows and columns.
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.
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.