
How many pieces can a puzzle have? - sfg
https://gottwurfelt.com/2020/08/17/how-many-pieces-can-a-puzzle-have/
======
robinhouston
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!

[https://www.puzzlemaster.ca/browse/novelty/packing/12838-jig...](https://www.puzzlemaster.ca/browse/novelty/packing/12838-jigsaw-
puzzle-19)

~~~
jacquesm
Nice but _super_ simple, the tips are unique which makes it easy to solve.
Still, very original.

~~~
_ZeD_
not _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)

------
cdubzzz
Somewhat 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-
puzzle...](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)

~~~
Xavdidtheshadow
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.

[0]: [https://fourpointpuzzles.com/products/the-
moon](https://fourpointpuzzles.com/products/the-moon)

~~~
cdubzzz
Wow, 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.

------
irrational
> 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
> 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.

~~~
irrational
My kids are working on a puzzle right now. I walked over and took a picture of
a part of the puzzle:

[https://imgur.com/rUrTPC3](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.

~~~
te
Who is the manufacturer? I don't think I've ever seen a non-grid jigsaw puzzle
before.

~~~
irrational
Bits and Pieces.

But we have hundreds of puzzles from tons of manufacturers. Other than the
puzzles for little kids, all of our puzzles are non-grids.

------
haar
Somewhat 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__
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.

Highly recommended.

------
pansa2
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.

~~~
vitus
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:

    
    
        --X---X--
        X---X---X
        --X---X--
        X---X---X

~~~
mahathu
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](https://i.imgur.com/T5yh9.jpg)

------
dhosek
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](https://puzzlemontage.crevado.com)

~~~
dhosek
I 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-
jigs...](https://www.nytimes.com/2020/04/08/business/coronavirus-jigsaw-
puzzles.html)

------
ConsiderCrying
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.

~~~
wjnc
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.)

~~~
foobarian
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.

~~~
skipnup
The last puzzle I solved actually had a small table with the advertised and
actual number of pieces.

------
dmurray
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.

~~~
beervirus
He was allowing for aspect ratios all the way out to 2:1 (which seems too long
and skinny to me).

~~~
messe
That's for the entire puzzle. I think they meant that the pieces themselves
have closer to 3:2 aspect ratios.

~~~
beervirus
Oops, I misread.

------
dehrmann
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.

~~~
jayd16
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.

------
__s
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?

~~~
Tokkemon
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.

------
ferros
Puzzles are a great way to pass that Corona time at home.

------
eru
The number theory aspects here are interesting. In practice, a puzzle that
slightly cheats the grid might also be interesting.

~~~
kuon
My 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:)

~~~
rcxdude
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).

~~~
eru
Might 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.

------
rcgorton
The 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.

------
staz
just realised if you want "squarish" piece you can't really have a rectangular
puzzle with a prime number of pieces without resorting to tricks

------
david_draco
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.

~~~
pfortuny
Why? Are you not missing the “interior”?

~~~
tgb
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.

~~~
bo1024
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.

------
mauvehaus
You 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
Is the box labelled "Almost 200 pieces"?

------
noja
Stupid question: why does a puzzle follow a grid?

~~~
__MatrixMan__
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/](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.

------
lepton
Ah, but how many sequences of 3 evenly spaced numbers does the sequence have?

