# 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):

 3 21 14 10 20 4
 33 18 9 6 11 22
 4 10 12 15 6 5
 2 3 4 108 54 36 27
 1 24 8 12 5 15 10
 1 63 224 42 192 54 288
 2 28 5 25 21 9 30

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

Solutions were sent by Joe DeVincentis, Lewis Chen, and Johannes Waldmann.

Here are the known shapes of SPP's with the smallest known entries:

6
 4 10 12 15 6 5

26 / 60
 33 18 9 6 11 22

33 / 198
 3 21 14 10 20 4

24 / 840

7
 2 3 4 108 54 36 27

117 / 108
 1 24 8 12 5 15 10

25 / 120
 1 63 224 42 192 54 288

288 / 12,096
 2 28 5 25 21 9 30

30 / 6300
 60 40 20 10 50 12 48

60 / 2400 (JD)

8
 2 10 12 20 30 6 5 3

44 / 60
 30 10 15 3 2 6 4 20

30 / 60
 20 5 6 4 3 12 10 15

25 / 60
 12 1 16 4 10 15 5 24

29 / 240
 1 10 5 6 8 3 9 2

11 / 360
 500 70 567 3 30 540 45 525

570 / 283500 (LC)
 96 60 36 1 5 90 32 64

96 / 5760 (JD)

9
 2 3 4 6 60 30 20 15 10

75 / 60
 90 8 16 30 36 1 45 24 20

90 / 720 (JD)
 5 10 30 20 1 18 6 9 36

45 / 180
 8 32 18 6 36 16 48 1 9

58 / 288
 1 10 15 12 6 8 20 4 2

26 / 240
 16 14 1 21 8 24 6 28 2

30 / 672
 24 12 1 21 14 28 2 6 36

36 / 1008
 6 40 2 8 36 45 1 16 30

46 / 1440
 4 36 8 32 15 25 24 16 40

40 / 460,800

10
 2 8 12 30 40 60 15 10 4 3

92 / 120
 27 18 12 2 36 4 6 9 54 3

57 / 108
 1 6 12 20 5 10 15 9 36 3

39 / 180
 60 1 8 12 2 10 45 30 3 36

69 / 360
 24 1 3 15 5 6 10 8 20 4

24 / 120
 24 15 4 5 6 8 10 20 1 3

24 / 120
 3 12 15 4 10 16 2 8 20 30

30 / 240
 28 15 10 3 2 12 14 1 7 20

28 / 840
 24 4 10 2 16 12 1 15 8 20

28 / 960

SPP's do not have to be connected. Are these the smallest possible product and sum?

 2 4 7 24 40 70 105 140 168 280 420 210 120 35 21 12 8 6 5 3 840

840 / 840 (JD)
 9 40 8 10 2 45 24 27 6 18 36 3 30 12 15 4 5 48

57 / 2160 (JW)

Theorem: If a SPP has a singleton column, it must have at least 3 other columns.
Proof: Say a column contains only the number n. Since the numbers must all be different, and each column has product n, the two largest numbers besides n are at most n/2 and n/3, which total less than n. (Joe DeVincentis points out this also proves that every other row besides the singleton column must contain at least 3 entries.)

Theorem: A SPP cannot contain singleton column and a singleton row unless it is the same singleton.
Proof: If a row only contains the number n, then because the row sums are constant, every other number in the SPP is smaller than n. This means the column that only contains one number cannot have product at least n.

Theorem: A SPP cannot contain two singleton columns or two singleton rows.
Proof: This follows directly from the fact that all the numbers need to be different.

Theorem: No SPP can have exactly two numbers in each row and each column.
Proof: (by Joe DeVincentis) Suppose we have a SPP {{A,S–A,,...},{,B,S–B,...},{,,C,S–C,...},...}, and assume without loss of generality that B>C. Their common product implies B(S–A) = C(S–B), or B/C = (S–B)/(S–A). This means S–B>S–A which implies A>B, and working around the chain we reach a contradiction.

Theorem: The only SPP with fewer than 6 numbers is the trivial 1×1 SPP.
Proof: Only a 2×2 SPP avoids a a singleton row or column, and that is impossible by the previous theorem. If there is a singleton column, there must be at least 6 other entries. The only remaining case is {{a,c–a},{b,c–b},{c,}}. But abc=(c–a)(c–b) or abc=c2–ac–bc+ab implies that ab divides c, so ab≥c, so abc≥c2>(c–a)(c–b).

Unknown Cases
7 8 9 (among others)

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 10/1/18.