Imagine a 24-by-24 chessboard. Now suppose you started counting all of the "sub-squares" on that board, squares of lengths 1 through 24 found by tracing the sides of the squares of the big board. To remind you how many sub-squares you've counted, you make a pile of little squares of all equal size (which you just happen to have lying around), one little square for each sub-square.

It turns out that these little squares can be put together, edge to edge, to form an even bigger chessboard.

What is the length of each side of the giant chessboard?

There can be squares that start at the full size of the 24x24 board, 23x23, 22x22, etc. down to 1x1. There's only one 24x24 square--that's the whole thing. With a 23x23 square, the upper left corner can be placed at any of 2x2 intersections (including the top left of the original board). A 22x22 square can be placed in 3x3 positions, etc.

These add up as shown in the following table:

 1 (24)    1     1  1.0000


 2 (23)    4     5  2.2361


 3 (22)    9    14  3.7417


 4 (21)   16    30  5.4772


 5 (20)   25    55  7.4162


 6 (19)   36    91  9.5394


 7 (18)   49   140 11.8322


 8 (17)   64   204 14.2829


 9 (16)   81   285 16.8819


10 (15)  100   385 19.6214


11 (14)  121   506 22.4944


12 (13)  144   650 25.4951


13 (12)  169   819 28.6182


14 (11)  196  1015 31.8591


15 (10)  225  1240 35.2136


16 ( 9)  256  1496 38.6782


17 ( 8)  289  1785 42.2493


18 ( 7)  324  2109 45.9239


19 ( 6)  361  2470 49.6991


20 ( 5)  400  2870 53.5724


21 ( 4)  441  3311 57.5413


22 ( 3)  484  3795 61.6036


23 ( 2)  529  4324 65.7571


24 ( 1)  576  4900 70.0000

Which shows the square size in parentheses.

These are listed by the size of the square of possible placement, the total possibilities for that size, the total thus far and that total's square root.

The first time there is a perfect square is indeed at line 24, so a size 24 board to begin with is the smallest for which this will work, and the final board is 70x70.

Posted by Charlie on 2003-06-30 03:20:17