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.

321
1410
204
41012
1565
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.

247244070105140168280
4202101203521128653

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 4x4 SP matrices with small entries.

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

Smallest Known Sum-Product Matrices

 
41012
1565
2101220
30653
28123040
60151043
1612152030
60105432
181020244060
12015126532
321
1410
204
11015
1268
2042
191215
185104
20836
1391836
24162052
30154810
149103545
20212818314
631557122
14816202456
402830514210
42157216353
110
56
83
92
1225
9154
10126
2053
191520
102483
165186
274212

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.