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".
 A: 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.
A: 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.
