I cut myself a rectangle out of a chessboard. (All cuts were along the square lines, so the rectangle has integer sides.)

Were I to tell you either its area, or its perimeter, or the length of its diagonal, you wouldn't be able to determine the dimensions of the rectangle.

What's its size?

The way I did this was to run a small program to find all the possible diagonals from cut rectangles:

DEFDBL A-Z
OPEN "cutchess.txt" FOR OUTPUT AS #1
FOR i = 1 TO 8
 FOR j = i TO 8
  PRINT #1, SQR(i * i + j * j), i, j
 NEXT
NEXT
CLOSE

The output was then sorted by diagonal length, and it was found that only 7.07106 and 8.062257 were shared by more than one pair of dimensions: the former by 1x7 and 5x5 and the latter by 1x8 and 4x7.

All of these have non-unique perimeters, so perimeter doesn't narrow it down any further.

But 1x7 has a unique area as 7 is prime. Likewise 5 is prime, so the only other way to get 25 would be 1x25 and the chessboard is not that big. The 28 area from 4x7 is also unique, as the only alternatives, 1x28 and 2x14 are too big for the chessboard.

That leaves 1x8 as having the only ambiguous area as area 8 can also be 2x4.

So the answer is 1x8.

Posted by Charlie on 2003-09-26 10:03:32