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Subject: Possible scores and factoring rss

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Russ Williams
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We've been playing lately and noting our scores, so I got interested in which scores are ambiguous, e.g. a score of 60 could be from having groups of size 5 and 12 or from groups of size 6 and 10.


1 2 3 4 5 6 7 8 9 10 11 12 13 14
1 1 2 3 4 5 6 7 8 9 10 11 12 13 14
2 4 6 8 10 12 14 16 18 20 22 24 26 28
3 9 12 15 18 21 24 27 30 33 36 39 42
4 16 20 24 28 32 36 40 44 48 52 56
5 25 30 35 40 45 50 55 60 65 70
6 36 42 48 54 60 66 72 78 84
7 49 56 63 70 77 84 91 98
8 64 72 80 88 96 104 112
9 81 90 99 108 117 126
10 100 110 120 130 140
11 121 132 143 154
12 144 156 168
13 169 182
14 196


Thus (unless I've overlooked something) the ambiguous cases are:

4 = 1*4 = 2*2
6 = 1*6 = 2*3
8 = 1*8 = 2*4
9 = 1*9 = 3*3
10 = 1*10 = 2*5
12 = 1*12 = 2*6 = 3*4
14 = 1*14 = 2*7
16 = 2*8 = 4*4
18 = 2*9 = 3*6
20 = 2*10 = 4*5
24 = 2*12 = 3*8 = 4*6
28 = 2*14 = 4*7
30 = 3*10 = 5*6
36 = 3*12 = 4*9 = 6*6
40 = 4*10 = 5*8
42 = 3*14 = 6*7
48 = 4*12 = 6*8
56 = 4*14 = 7*8
60 = 5*12 = 6*10
70 = 5*14 = 7*10
72 = 6*12 = 8*9
84 = 6*14 = 7*12

Only 12, 24 and 36 are ambiguous with 3 possible factorizations instead of 2 possible factorizations.

In practice, if players are competent and trying to win, I don't think any of the ambiguities are important: scores almost always seem to be 50 or higher, often 90 or more, and the 2 groups seem to usually have reasonably close sizes.

E.g. for a recorded score of 70, I'd certainly expect 7*10 rather than 5*14.
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