Imagine a 24by24 chessboard. Now suppose you started counting all of the "subsquares" 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 subsquares 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 subsquare.
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 squarethat'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 20030630 03:20:17 