# How to calculate the number of pieces in the border of a puzzle?

Is there any way to calculate how many border-pieces a puzzle has, without knowing its width-height ratio? I guess it's not even possible, but I am trying to be sure about it.

BTW you might want to know that the puzzle has 3000 pieces.

• Assuming the puzzle is some kind of Rectangle or a Square, you can find the number of border pieces only if you know the size of the full puzzle as well as the size of the small pieces(assuming all small pieces are of the same size).
– lsp
Commented Mar 22, 2013 at 11:39
• The problem is, you can't even be sure your puzzle has 3000 pieces. It could very well have 3008, for example, but be advertised as 3000. And let's hope you're speaking of a puzzle with regular square-shaped pieces with nothing fancy Commented Mar 22, 2013 at 13:00
• @Joubarc The last reference is just brilliant! :-) Commented Mar 22, 2013 at 13:04
• The Wikipedia page has more examples, but I found that one the most impressive. Its border pieces are also a nice illustration of ways puzzle makers can mess with the part count. Commented Mar 22, 2013 at 13:09
• As @Joubarc: says most puzzles I have seen have somewhat more pieces than the number on the box. But roughly speaking $4 \sqrt {\text{pieces}}$ is a lower limit and the real answer won't be too much greater. Commented Mar 22, 2013 at 13:17

Obviously, $w\cdot h=3000$, and there are $2w+h-2+h-2=2w+2h-4$ border pieces. Since $3000=2^3\cdot 3\cdot 5^3$, possibilities are \begin{eqnarray}(w,h)&\in&\{(1,3000),(2,1500),(3,1000),(4,750),(5,600),(6,500),\\&&\hphantom{\{}(8,375),(10,300),(12,250),(15,200),(20,150),(24,125)\\ &&\hphantom{\{}(25,120),(30,100),(40,75),(50,60),(h,w)\},\end{eqnarray}

Considering this, your puzzle is probably $50\cdot60$ (I've never seen a puzzle with $h/w$ or $w/h$ ratio more than $1/2$), so there are $216$ border pieces. This is only $\frac{216\cdot100\%}{3000}=7.2\%$ of the puzzle pieces, which fits standards.

Admitting it's a regular grid of 3000 pieces, no more, no less, there aren't that many possibilities for the size of the borders, as they must be a divisor of 3000.

However, even if you don't know the ratio, it's usually safe to assume it's between 1:1 and 2:1, as most puzzles are nice rectangles (using a very-non-mathematical notion of nice, by which I mean, most pictures, paintings, sheets of paper, screens, etc...).

This leaves us with the following possibilities:

• 50x60 -> 2x48 + 2x58 = 212 border pieces and 4 corners
• 40x75 -> 2x38 + 2x74 = 224 border pieces and 4 corners
• 30x100 is already far from a 2:1 ratio but would yield 2x28 + 2x98 = 252 border pieces and 4 corners.

Assume that the puzzle is a rectangle of $a\times b\$ cm$^2$, and the pieces can be idealized as rectangles of $c\times d\$ cm$^2$. Then $m={a\over c}$ pieces border along an $a$-side and $n={b\over d}$ pieces along a $b$-side. You have told us that $m\cdot n=3000$, and you want to know the number $N:=2m+2n-4$. We don't have enough information to determine $N$. By the AGM-inequality one has ${m+n\over 2}\geq \sqrt{mn}$. This implies the estimate $N\geq216$.

Think of this like a perfect square. Example say the puzzle is 100 pieces, that's 10X10 grid, so that puzzle would have 40 boarder pieces approximately. There is no way to figure exactly the width and height. So going by that using a 300X300 grid equaling 3000 pieces, you best answer is approximately 1200 for all four sides of the boarder. If you want a realistic answer, just Google some 3000 piece puzzles and find many similar ones then get your best answer.

You could apply the Monte Carlo method. Pick a random piece (that is, with an uniform distribution) and note if it is a border piece or not. Repeat until the ratio $$k = \frac{\# \text{noticed border pieces}}{\#\text{total}}$$ converges (up to the desired precision). Then $3000 \cdot k$ will be your answer (or generally $nk$ for $n$-piece puzzle).

Good luck! ;-)

• I really want to vote this up, but can't in good conscience since it's not (meant to be) helpful. :-) The other answers aren't all that helpful either, due to the issues raised in the comment. Commented Mar 22, 2013 at 14:30

500 piece has 88 border pieces. 750 piece has 108 border pieces. 1000 piece has 128 border pieces. These numbers hold true for standard rectangle puzzles.

• These all assume the number of pieces is exactly correct and the puzzle is as square as possible. Both assumptions are suspect. Commented Jul 7, 2016 at 2:35