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?

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) 126 420 420 26 6 100 360 364 182 546 / 52920 (GS)

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) 27 20 2 5 15 12 18 9 27 / 540 (GS) 30 6 9 15 20 10 12 18 30 / 180 (GS)

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

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).

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.