# Jigsaw Puzzle Help

I have a puzzle I'm trying to solve and I have all the border pieces set out but I'm pretty sure some are missing. How do I figure out how many pieces are in the border? It's a 1000 piece puzzle and the dimensions are 20" by 27".

• The first and easiest thing to check is if you have an even or odd number of border pieces. Sep 1, 2013 at 21:05
• $20''\times 27'' = 540$ sq inches Equally dividing that over 1000 pieces gives us $0.54$ sq inches per piece. Assuming that the pieces are square, we have a borderlength of $\sqrt{0.54}=0.735''$ per piece. The amount of pieces along one border is $27/0.735=36.73$ - This is not even close to an integer, meaning our assumption (pieces are square and equally in size) was incorrect - we need to know that in order to give qualified answers ;) Sep 1, 2013 at 21:07
• Because puzzles don't necessarily require all pieces to be in nice orderly rows/columns, I believe the easiest way to do this is to actually construct the puzzle. :) Sep 1, 2013 at 21:07
• The answer can only be approximative and depends on the average aspect ratio of the pieces. When this ratio is the same as for the whole puzzle the number of boundary pieces comes to about $2a+2(a-1)\doteq124.5$, where $a:=\sqrt{1000}$. Sep 1, 2013 at 21:07

Jigsaw puzzles almost never have the exact number of pieces that it says on the box, because the pieces tend to form a rectangle with all the individual pieces being rectangles of the same size. You can verify this by counting your own jigsaw puzzles. Note that a puzzle which claims to have $1000$ pieces will never have less than $1000$ but almost certainly has more.
If your puzzle happens to be a high-quality one with funny-shaped pieces, then it's possible that it does have exactly $1000$ pieces, in which case I don't know how to answer your question. But if it's a standard puzzle, it is very likely that it has $1026$ pieces in total (or would have, if there were none missing) and they form a $38$ by $27$ rectangle. This is because this is the standard size for "1000-piece" puzzles. In this case, the number of pieces around the edge is $$38 \times 2 + 27 \times 2 - 4$$ or $126$ in total. By the way, since your puzzle has a ratio of side lengths of $27/20$ and this is roughly $38/27$, I think it is very likely that your puzzle is of this standard size. You can check further by counting the number of pieces along the sides which you do have.
If we assume that the number of pieces is exactly 1000, then we are looking for a divisor $d$ of $1000$ such that $d:\frac{1000}d\approx 20:27$. The closest approximation would obtained with $d=25$ ($25:40\approx 20:27$), so there'd be $4$ corner pieces and $2\cdot(25-2)+2\cdot(40-2)= 122$ other edge pieces. But it is well possible that the number of total pieces is only approximately 1000 ...
With $999$ pieces, the same calculation arrives at $27:37\approx 20:27$ (where the approximation is much closer than above), so with this setup there'd be $4$ corner pieces and $2\cdot(27-2)+2\cdot(37-2)= 120$ other edge pieces.