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Problem of the Month (October 2018)

In the November 2002 Math Magic, we studied matrices of distinct positive integers with the property that each row had constant sum and each column had constant product. This month we generalize to allow some entries to be blank. We call any such shape a sum-product polyominoes or SPP. Below are the only known SPP's with 6 or 7 entries (up to permutations of rows and columns). We show them labeled with the smallest possible entries (by sum or product):

321
1410
204
33
1896
1122
41012
1565
234108
543627
124
8125
1510
163224
4219254
288
228
525
219
30

What shapes can a SPP with 8 or more entries have? What are the smallest possible entries?


You can see all the best known results here.

Submit your answers here.

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