Problem of the Month (June 2007)

Consider the equation n 12 + (n-1) 22 + (n-2) 32 + . . . + 1 n2 = k2. This equation holds for certain integer values of n and k. The first few solutions are n=1 k=1, n=6 k=14, and n=25 k=195. Can you find all the other solutions?

We can illustrate this identity for a shape S by tiling a copy of S scaled by a factor of k with: n copies of S, (n-1) copies of S scaled by a factor of 2, ... up to 1 copy of S scaled by a factor of n. These are called anti-partridge tilings. Clearly the n=1 k=1 case is trivial. Anti-partridge tilings for n=6 of squares and equilateral triangles were found a decade ago. These can be stretched to yield anti-partridge tilings for n=6 of all parallelograms and triangles. What anti-partridge tilings can you find for n=6 or n=25?

Some rare shapes can have anti-partridge tilings for other values of n. There are apparently rectangle solutions for all values of n. Can you find some? There are also solutions for n=4 and n=10 using triangles. Can you find them? What other anti-partridge tilings can you find for non-integer values of k?

ANSWERS

Joe DeVincentis noted that n 12 + (n-1) 22 + (n-2) 32 + . . . + 1 n2 = n(n+1)2(n+2)/12, so we want n so that 3n(n+2) is a square. The n with this property are of the form [ (2 - √3)n + (2 + √3)n - 2 ] / 2.

Patrick Hamlyn claimed to find all the solutions for the 30-60-90 triangle for n=6, but George Sicherman found many more.

Here are the known non-trivial anti-partridge tilings for n=6:

(Colin Singleton, 1996)

(Bob Wainwright, 2000)

(Erich Friedman, 2002)

(Patrick Hamlyn, 2002)

(Erich Friedman, 2002)

(Erich Friedman, 2002)

(Patrick Hamlyn, 2002)

(Patrick Hamlyn)

(Patrick Hamlyn)

(George Sicherman)

And here are the known non-trivial anti-partridge tilings for n=25:

(Erich Friedman, 1998)

(Mike Reid)

George Sicherman found many rectangles for other values of n.

Joe DeVincentis showed there are rectangle solutions for all n not equal to 1 mod 4 by stacking squares of size n+1–i above those of size i (like n=2 through 4 below).

Here are the known non-trivial anti-partridge tilings for other values of n:


n=2
n=3
n=4
n=5


n=6
n=7
n=7
n=7


n=7
n=8
n=8
n=8


n=8
n=8
n=8


n=9
n=9
n=9


n=10
n=10


n=4


n=10

Patrick Hamlyn likes tiling things on other surfaces too. He found this anti-partridge tiling for the bent triomino on a Möbius strip:

Patrick Hamlyn also found anti-partridge-like tilings to illustrate the following identities:

(1) 132 + (2) 122 + . . . + (10) 42 = 552

(4) 102 + (5) 92 + . . . + (11) 32 = 482

(6) 72 + (7) 62 + . . . + (11) 22 = 322

(7) 82 + (8) 72 + . . . + (14) 12 = 422

Claudio Baiocchi found this anti-partridge-like tiling:

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 7/2/07.