Robert T. Wainright's Partridge Puzzle of order n is: pack 1 1-by-1 square, 2 2-by-2 squares, ..., n n-by-n squares into a larger square of size N-by-N, where N is the sum of the first n integers.
The sum of the areas of the smaller squares is always equal to the area of the larger square (N-by-N) because of a basic identity about cubes: the sum of the first n cubes is always equal to the square of the sum of the first n integers.
But just because the area sums work out doesn't mean that you can actually tile the larger square correctly. In fact, there are no solutions for n < 8. But n = 8 has several nice packings (n = 12 does as well, and maybe larger values of n?) and you can find them all with the script in this directory.
Run ./solve-partridge.sh 8 to show all of the solutions for n = 8. You'll have to wait a few minutes for
the first one to show up. You should see output like this at the terminal:
┌───┬───┬─────────┬─────────────┬─────────────┬─────────────┬───────────┐
│ 2│ 2│ │ │ │ │ │
├───┴───┤ │ │ │ │ │
│ │ │ │ │ │ │
│ │ 5│ │ │ │ │
│ 4├─────────┤ │ │ │ 6│
├───────┤ │ 7│ 7│ 7├───────────┤
│ │ ├─────────────┴─┬───────────┴───┬─────────┤ │
│ │ │ │ │ │ │
│ 4│ 5│ │ │ │ │
├─────┬─┴─────────┤ │ │ │ │
│ │ │ │ │ 5│ 6│
│ 3│ │ │ ├─────┬───┴───────────┤
├─────┤ │ │ │ │ │
│ │ │ 8│ 8│ 3│ │
│ 3│ 6├─┬───────────┬─┴─────┬─────────┴─────┤ │
├─────┴───┬───────┴─┤ │ │ │ │
│ │ │ │ │ │ │
│ │ │ │ 4│ │ │
│ │ │ ├───────┤ │ 8│
│ 5│ 5│ 6│ │ ├───────────────┤
├─────────┴─────┬───┴───────────┤ │ │ │
│ │ │ 4│ 8│ │
│ │ ├───────┴───┬───────────┤ │
│ │ │ │ │ │
│ │ │ │ │ │
│ │ │ │ │ │
│ │ │ │ │ 8│
│ 8│ 8│ 6│ 6├───────────────┤
├─────────────┬─┴───────────┬───┴─────────┬─┴───────────┤ │
│ │ │ │ │ │
│ │ │ │ │ │
│ │ │ │ │ │
│ │ │ │ │ │
│ │ │ │ │ │
│ 7│ 7│ 7│ 7│ 8│
└─────────────┴─────────────┴─────────────┴─────────────┴───────────────┘