A369891 -id:A369891 - cp's OEIS frontend

A081287 Excess area when consecutive squares of sizes 1 to n are packed into the smallest possible rectangle.

0, 1, 1, 5, 5, 8, 14, 6, 15, 20, 7, 17, 17, 20, 25, 16, 9, 30, 21, 20, 33, 27, 28, 28, 22, 29, 26, 35, 31, 31, 34, 35

Offset: 1

Ed Pegg Jr, Mar 16 2003

Restricted to packings with the squares aligned with the sides of the rectangle.

			Verified best rectangles > 5 are as follows:
 6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24  25 26 27  28  29  30  31  32
--------------------------------------------------------------------------------------
 9 11 14 15 15 19 23 22 23 23 28 39 31 47 34 38 39 64 56  43 70 74  63  81  51  95  85
11 14 15 20 27 27 29 38 45 55 54 46 69 53 85 88 98 68 88 129 89 94 123 106 186 110 135
Visual representations are at the Tightly Packed Squares link.

R. K. Guy, Unsolved Problems in Geometry, Section D4, has information about several related problems.
R. M. Kurchan (editor), Puzzle Fun, Number 18 (December 1997), pp. 9-10.

Jean-François Alcover, Mathematica script (after E. Pegg and R. Korf)
R. Ellard and Des MacHale, Packing Squares into Rectangles, The Mathematical Gazette, Vol. 96, No. 535 (March 2012), 1-18.
Eric Huang and Richard E. Korf, New improvements in optimal rectangle packing
Richard E. Korf, Optimal Rectangle Packing: New Results, ICAPS, 2004.
Ed Pegg Jr, Square Packing
E. Pegg and R. Korf, Tightly Packed Squares.

Cf. A000330, A038666, A369891.

a(n) = A038666(n) - A000330(n). - Pontus von Brömssen, Mar 01 2024

Four extra terms computed by Korf, May 24 2005

More terms from Ed Pegg Jr, Feb 14 2008 and again Sep 16 2009

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

Original entry on oeis.org

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

Offset: 1

Danila Potapov, Mar 11 2025

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.

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.

Cf. A369891.

cp's OEIS Frontend

A081287 Excess area when consecutive squares of sizes 1 to n are packed into the smallest possible rectangle.

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