Monday s Solution

Monday's Solution

"Monday started out systematically. First, he worked with the number of ways a total could be rolled, knowing that we can always convert to probabilities by dividing the number of ways by 216. That, it turns out, is easier to think about than decimal numbers.

"His first approach," continued Roscoe, "was to see if all of the first nine 'ways' could be constructed. He came up with Table 20.5.

Table 20.5. How to Get the First Nine Ways

Ways	Need to Roll
1	3
2	3 or 18
3	4
4	3 or 4
5	3 or 4 or 18
6	5
7	3 or 5
8	3 or 5 or 18
9	4 or 5

"I'm beginning to see," I said. "What Monday is going to try to do is see if he can cover the entire set of 'ways' up to 108, using the elements in the 'ways' column of 1, 3, 6, 10, 15, 21, 25, and 27. Very clever."

"Yes," replied Roscoe, "that's the idea. But remember, he can only use each element twice. The symmetry of the problem is helpful, but notice that 'ways' get used up, so we need to be careful. As an example, suppose we used 3 and 18, and needed one more 'way.' We would be stuck. So there are constraints we have to watch out for."

"Wow," I said, "so the answer is still in doubt."

"Monday was equal to the task, it turns out," said Roscoe. "Here is the rest of his logic: To get 10 ways, you just use a roll of 6. By using 6 and its companion of 15, we have two '10s' to work with, so we can now get from 1 to 29. We can now add 1 to 29 to anything, provided we don't need any additional totals of 3, 4, 5, 6, 15, 16, 17, or 18. We have basically used up the 'ways' elements corresponding to 1, 3, 6, and 10."

"Well, 29 is still a long way from 108," I said.

Roscoe completed Monday's solution. "If you now use one of the '21s,' say 8, you extend the range from 29 to 50. And you still have both '15s', both '25s', and both '27s' to work with. Using the second 21 extends the range from 50 to 71. The pair of '27s' takes us to 125, well past the 108 we needed. So it is possible."

"What you are saying, then," I responded, "is that every way is possible, so that we can uniformly cover the interval in steps of 1/216. That is pretty amazing. Did you actually construct the table?"

"It was easy, once we knew it could be done," said Roscoe. "Table 20.6 is the final result."

Table 20.6. The Solution to the Problem

Ways	Probability	Roll Any of These Totals to Get This Probability
1	0.00463	3
2	0.00926	3	18
3	0.01389	4
4	0.01852	3	4
5	0.02315	3	4	18
6	0.02778	5
7	0.03241	3	5
8	0.03704	3	5	18
9	0.04167	4	5
10	0.04630	6
11	0.05093	3	6
12	0.05556	3	6	18
13	0.06019	4	6
14	0.06481	3	4	6
15	0.06944	7
16	0.07407	3	7
17	0.07870	3	7	18
18	0.08333	4	7
19	0.08796	3	4	7
20	0.09259	6	15
21	0.09722	8
22	0.10185	3	8
23	0.10648	3	8	18
24	0.11111	4	8
25	0.11574	9
26	0.12037	3	9
27	0.12500	10
28	0.12963	3	10
29	0.13426	3	10	18
30	0.13889	4	10
31	0.14352	6	8
32	0.14815	3	6	8
33	0.15278	3	6	8	18
34	0.15741	4	6	8
35	0.16204	6	9
36	0.16667	3	6	9
37	0.17130	3	6	9	18
38	0.17593	3	6	10
39	0.18056	3	6	10	18
40	0.18519	7	9
41	0.18981	3	7	9
42	0.19444	8	13
43	0.19907	3	8	13
44	0.20370	3	8	13	18
45	0.20833	6	9	15
46	0.21296	8	9
47	0.21759	3	8	9
48	0.22222	8	10
49	0.22685	3	8	10
50	0.23148	9	12
51	0.23611	3	9	12
52	0.24074	9	10
53	0.24537	3	9	10
54	0.25000	10	11
55	0.25463	3	10	11
56	0.25926	3	10	11	18
57	0.26389	4	10	11
58	0.26852	6	8	10
59	0.27315	3	6	8	10
60	0.27778	5	10	11
61	0.28241	3	5	10	11
62	0.28704	6	9	10
63	0.29167	3	6	9	10
64	0.29630	6	10	11
65	0.30093	3	6	10	11
66	0.30556	3	7	9	12
67	0.31019	7	9	10
68	0.31481	3	7	9	10
69	0.31944	7	10	11
70	0.32407	3	7	10	11
71	0.32870	8	9	12
72	0.33333	3	8	9	12
73	0.33796	3	8	9	12	18
74	0.34259	3	8	9	10
75	0.34722	8	10	11
76	0.35185	3	8	10	11
77	0.35648	9	10	12
78	0.36111	3	9	10	12
79	0.36574	9	10	11
80	0.37037	7	9	12	14
81	0.37500	3	7	9	12	14
82	0.37963	3	7	9	12	14	18
83	0.38426	3	4	9	10	11
84	0.38889	7	10	11	14
85	0.39352	3	7	10	11	14
86	0.39815	3	7	10	11	14	18
87	0.40278	4	7	10	11	14
88	0.40741	3	4	7	10	11	14
89	0.41204	6	9	10	11
90	0.41667	3	6	9	10	11
91	0.42130	3	6	9	10	11	18
92	0.42593	8	9	12	13
93	0.43056	3	8	9	12	13
94	0.43519	3	8	9	12	13	18
95	0.43981	4	8	9	12	13
96	0.44444	8	10	11	13
97	0.44907	3	8	10	11	13
98	0.45370	3	8	10	11	13	18
99	0.45833	4	8	10	11	13
100	0.46296	6	7	9	12	14	15
101	0.46759	3	8	9	10	11
102	0.47222	4	8	10	11	13	17
103	0.47685	3	5	8	10	11	13
104	0.48148	9	10	11	12
105	0.48611	3	9	10	11	12
106	0.49074	3	9	10	11	12	18
107	0.49537	4	9	10	11	12
108	0.50000	3	4	9	10	11	12

"Note," said Roscoe, "that for some of these the answer is not unique. But remember also that it doesn't have to be. We just need to find one set of totals that gives us the right number of 'ways' to get there."