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Only 19, 21 and 23 would fail to be able to produce 188, so it must be one of these. The table shows one set of coefficients that works to produce 188, for each possible date of Carol's birth.
188:
coefficient of
c 19 21 c
1 0 0 188
2 0 0 94
3 2 0 50
4 0 0 47
5 0 3 25
6 2 0 25
7 4 0 16
8 0 4 13
9 2 2 12
10 0 8 2
11 1 7 2
12 2 2 9
13 0 4 8
14 4 0 8
15 2 0 10
16 1 5 4
17 1 4 5
18 2 2 6
19
20 0 8 1
21
22 1 7 1
23
24 2 6 1
25 0 3 5
26 0 4 4
27 2 2 4
28 4 0 4
29 2 3 3
30 2 0 5
31 0 6 2
Any date for Carol's birth would allow 189 to be formed:
189:
c 19 21 c
1 0 0 189
2 0 1 84
3 0 0 63
4 0 1 42
5 0 4 21
6 0 1 28
7 0 0 27
8 0 1 21
9 0 0 21
10 0 9 0
11 0 9 0
12 0 1 14
13 0 9 0
14 0 1 12
15 0 4 7
16 0 9 0
17 0 9 0
18 0 3 7
19 0 9 0
20 0 9 0
21 0 0 9
22 0 9 0
23 0 9 0
24 0 1 7
25 0 9 0
26 0 9 0
27 0 0 7
28 0 1 6
29 0 9 0
30 0 9 0
31 0 9 0
However, 19 and 21, which, by the way, repeat Alice or Becky's birthday, fail to allow 191 to be formed:
191:
c 19 21 c
1 : 0 0 191
2 : 0 1 85
3 : 2 0 51
4 : 0 3 32
5 : 0 1 34
6 : 2 1 22
7 : 6 0 11
8 : 0 3 16
9 : 2 0 17
10 : 0 1 17
11 : 0 7 4
12 : 2 1 11
13 : 0 6 5
14 : 6 1 4
15 : 2 3 6
16 : 0 3 8
17 : 0 1 10
18 : 2 3 5
19 :
20 : 9 0 1
21 :
22 : 0 7 2
23 : 0 8 1
24 : 2 5 2
25 : 1 7 1
26 : 1 2 5
27 : 2 6 1
28 : 6 1 2
29 : 3 5 1
30 : 2 3 3
31 : 4 4 1
That leaves 23 as the only possible date for Carol's birthday.
From Enigma No. 1635, "Marty's date", by Gwen Owen, New Scientist 26 February 2011, page 29.
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