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 sumproduct matrices, or SP matrices.
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?
ANSWERS
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 SumProduct Matrices









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  9  15  20 
10  24  8  3 
16  5  18  6 
27  4  2  12 

If you can extend any of these results, please
email me.
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