# Problem of the Month (November 2002)

The matrices below are interesting, because they both have the properties that 1) all of their entries are distinct positive integers, 2) the sum of each row is the same, and 3) the product of each column is the same. We call these matrices sum-product matrices, or SP matrices.

 3 21 14 10 20 4
 4 10 12 15 6 5
In fact, these are the 3×2 and 2×3 SP matrices with the smallest entries. It is easy to see that there are no n×1, 1×n, or 2×2 SP matrices. Do they exist in all other sizes? What are the smallest SP matrices of other sizes? Are there any SP matrices where the row sums and the column products are equal?

Joseph DeVincentis found the smallest 2×n SP matrices up to 2×16, the smallest 3×n SP matrices up to 3×7, and plenty of other small SP matrices.

Joseph DeVincentis also found some SP matrices with the same sum and product. The smallest one possible (shown below) has row sum and column product 840. He also conjectures that all matrices that have this property are 2×n.

 2 4 7 24 40 70 105 140 168 280 420 210 120 35 21 12 8 6 5 3

Both Philippe Fondanaiche and Brendan Owen found proofs that SP matrices of all other sizes exist. Both proofs are pretty messy.

Philippe Fondanaiche also found some small SP matrices. Carlos Ungil found many 4×4 SP matrices with small entries.

Here are the smallest known SP matrices of a given size:

## Smallest Known Sum-Product Matrices

 4 10 12 15 6 5
 2 10 12 20 30 6 5 3
 2 8 12 30 40 60 15 10 4 3
 1 6 12 15 20 30 60 10 5 4 3 2
 1 8 10 20 24 40 60 120 15 12 6 5 3 2
 3 21 14 10 20 4
 1 10 15 12 6 8 20 4 2
 1 9 12 15 18 5 10 4 20 8 3 6
 1 3 9 18 36 24 16 20 5 2 30 15 4 8 10
 1 4 9 10 35 45 20 21 28 18 3 14 63 15 5 7 12 2
 1 4 8 16 20 24 56 40 28 30 5 14 2 10 42 15 7 21 6 35 3
 1 10 5 6 8 3 9 2
 1 2 25 9 15 4 10 12 6 20 5 3
 1 9 15 20 10 24 8 3 16 5 18 6 27 4 2 12

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 11/20/02.