OFFSET
1,8
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.
For any order-n solution with n being even, an order-(n+1) solution can be formed by putting the (n+1) X (n+1) squares in an L-shape around the order-n solution. Since a(8) = 2332, there are 4 * 8 * 2332 = 74624 L-shape (non-distinct) solutions in the Matt Parker video and not 352072 as claimed. - Ruediger Jehn, Sep 11 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).
Oleg Tarakanov, Partridge Solver.
Robert T. Wainwright, The Partridge Puzzle.
FORMULA
a(n) = 0 if and only if A369891(n) > 0. - Pontus von Brömssen, Dec 09 2025
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Danila Potapov, Mar 11 2025
EXTENSIONS
a(10) from Oleg Tarakanov, Dec 03 2025
STATUS
approved