Since 1^2 + 2^2 + 3^2 + ...

geometry/dissections/square.70

Problem

Since 1^2 + 2^2 + 3^2 + ... + 24^2 = 70^2, can a 70x70 square be dissected into
24 squares of size 1x1, 2x2, 3x3, etc.?

Show solution

Martin Gardner asked this in his Mathematical Games column in the
September 1966 issue of Scientific American.  William Cutler was the first
of 24 readers who reduced the uncovered area to 49, using all but the 7x7
square.  All the patterns were the same except for interchanging the
squares of orders 17 and 18 and rearranging the squares of orders 1, ...,
6, 8, 9, and 10.  Nobody proved that the solution is minimal.

+----------------+-------------+----------------------+---------------------+
|                |             |                      |                     |
|                |             |                      |                     |
|                |      11     |                      |                     |
|                |             |                      |                     |
|       16       |             |                      |                     |
|                +-----+--+----+         22           |         21          |
|                |     | 2|    |                      |                     |
|                |  5  +--+----+                      |                     |
|                |     |       |                      |                     |
+----------------+--+--+   6   |                      |                     |
|                   | 3|       |                      |                     |
|                   ++-+-------+                      |                     |
|                   ||         |                      ++--------------------+
|                   ||    8    +----------------------++                    |
|        18         ||         |                       |                    |
|                   ||         |                       |                    |
|                   ++---------+                       |                    |
|                   |          |                       |         20         |
|                   |     9    |                       |                    |
+------------------++          |          23           |                    |
|                  ||          |                       |                    |
|                  ++----------+                       |                    |
|                  |           |                       +---++---------------+
|                  |           |                       |   ||               |
|        17        |     10    |                       | 4 ||               |
|                  |           +---------------+-------+---++               |
|                  +-+---------+---------------+            |      15       |
|                  | |                         |            |               |
|                  | |                         |     12     |               |
+------------------+-+                         |            +-+-------------+
|                    |                         |            |1|             |
|                    |                         +------------+-+             |
|                    |           24            |              |             |
|                    |                         |              |             |
|        19          |                         |     13       |     14      |
|                    |                         |              |             |
|                    |                         |              |             |
|                    |                         |              |             |
+--------------------+-------------------------+--------------+-------------+