## Ken's POTW

Sums of Squares
The current Macalaster Problem of the Week (#841) reads:
*
Find the first integer k for which 169 is not the sum of k nonzero squares.
*

The squares are not necessarily unique. For example 169=1+4+4+16+144.
Here are some extensions I think are also worth investigating:

- The above problem.
- For each k from 1 to 10 (and farther?), find the first integer
which is the sum of 1,2,...,k nonzero squares.
- Find the integer from 1 to 100 which can be decomposed into sums of
1,2,...,k nonzero squares, where k is a maximum.
- Find the integer from 1 to 100 which can be decomposed into the
most different sums of squares (not necessarily 1,2,...,k).

The Macalaster problem suggests that behind the decompositions is the
request for an efficient computer algorithm. If you find a good one,
please send it along.
Source:
Based upon
Macalester College Problems of the Week
#841.

Solution

Mail to Ken