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 3x2 and 2x3 SP matrices with the smallest entries. It is easy to see that there are no nx1, 1xn, or 2x2 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?

ANSWERS

Joseph DeVincentis found the smallest 2xn SP matrices up to 2x16, the smallest 3xn SP matrices up to 3x7, 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 2xn.

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.