# Is there a geometric realization in integer-sided squares of $70^2 =\sum_{j=1}^{24} j^2$?

I saw this in the NAdigest mailing list, and it was obviously suggested by $70^2 =\sum_{j=1}^{24} j^2$:

From: Gerhard Opfer gerhard.opfer@uni-hamburg.de

Date: November 06, 2017

Subject: Mathematics, combinatorial

Is it known, whether a square Q of size 70 x 70 can be covered by little squares q_j of size j x j, j=1,2,...,24.

I don't know.

My first thought was to look at the unit square. However, I realized that it was possible to surround the unit square with larger squares.

The fact that it is possible to square the square (i.e., fill an integer-sided square with distinct integer-sided squares - see https://en.wikipedia.org/wiki/Squaring_the_square) means that some property of 70 and 1 through 24 is needed if it is not possible.

It just might be impossible to square a 70 x 70 square.

• Well, it is one reason that the Leech Lattice works. see SPLAG by Conway and Sloane, or Lattices and Codes by Ebeling. Page 130 in the second edition of Ebeling. – Will Jagy Nov 8 '17 at 4:44
• It is impossible, see answers in this question on MO – achille hui Nov 8 '17 at 5:47

According to the wikipedia link you've posted, the smallest (in terms of side length) perfect squared squares are $110\times110$, so it appears no such realization is possible.
The teacher misses the fact that he can combine his solid with a center-inverted version of itself to form a true rectangular block, containing six stepped pyramids and having the dimensions $(n)×(n+1)×(2n+1)$. With this modification apply the procedure to $n=24$, thus forming a $24×25×49$ block. Divide this block into six equal layers by cutting orthogonally to the $24$ dimension. Each of these six layers $(4×25×49)$ has the same volume as one of the original stepped pyramids and also can be subdivided into a $2×5×7$ array of $2×5×7$ blocks.