A381976 a(n) is the number of distinct solutions to the Partridge Puzzle of size n.

1, 0, 0, 0, 0, 0, 0, 2332, 216285

Offset: 1

Author

Danila Potapov, Mar 11 2025

Keywords

• nonn
• more
• changed

Comments

a(n) is the number of packings of squares of side 1..n to fill the square of side n(n+1)/2 under the condition that there are: 1 square of size 1 X 1, 2 squares of size 2 X 2, 3 squares of size 3 X 3, ..., n squares of size n X n.

The sequence comes from the formula 1^3 + 2^3 + ... + n^3 = (1+2+...+n)^2 = (n(n+1)/2)^2 (Nicomachus's theorem), so that the areas of the squares sum up to the area of the big square.

Rotations and mirrorings of the packings are not counted as distinct (there are in total 8 distinct variations of each packing).

Interestingly, for n = 9 the area of the big square is equal to 45*45 = 2025 making this problem a problem of the year 2025.

Links

• Matt Parker, The impossible puzzle with over a million solutions!, YouTube video.
• Danila Potapov, How I solved a problem of the year 2025 (in Russian).
• Robert T. Wainwright, The Partridge Puzzle.

Crossrefs

Cf. A369891.