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 sumproduct 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):
What shapes can a SPP with 8 or more entries have? What are the smallest possible entries?
ANSWERS
Solutions were sent by Joe DeVincentis and Lewis Chen.
Here are the known shapes of SPP's with the smallest known entries:
6

26 / 60

33 / 198

24 / 840


7

117 / 108

25 / 120

288 / 12,096

30 / 6300

60 / 2400 (JD)


8

44 / 60

30 / 60

25 / 60

29 / 240

11 / 360

570 / 283500 (LC)

96 / 5760 (JD)


9

75 / 60

90 / 720 (JD)

45 / 180

58 / 288

26 / 240

30 / 672

36 / 1008

46 / 1440

40 / 460,800


10

92 / 120

57 / 108

39 / 180

69 / 360

24 / 120

24 / 120

30 / 240

28 / 840

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)

90  80  60  8    
 16  18  24  180    
     144  45  40  9
     10  32  36  160
 238 / 1440


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=c^{2}–ac–bc+ab implies that ab divides c, so ab≥c, so abc≥c^{2}>(c–a)(c–b).
Unknown Cases
If you can extend any of these results, please
email me.
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