You are playing a game where the caller rolls two dice and tells you the numbers, and you have to make the numbers on your card by adding, subtracting, multiplying or dividing the two numbers on the dice. You can make more than one number with each pair of dice; eg if the caller rolls a 6 and a 5 you could make 1, 11 & 30 in the same go.
What would be the best six positive integers to have on your card to have the best chance of winning?
The best numbers to have on your card are those that are more likely to have a way of being formed from numbers that come up as the dice totals. The probability of being able to form a given number is proportional to the number of ways the two dice can fall that can be used to form that number.
The number of ways that the dice can fall to produce given numbers is:
1 16
2 13
3 12
4 9
5 8
6 9
7 6
8 7
9 5
10 5
11 2
12 5
15 2
16 1
18 2
20 2
24 2
25 1
30 2
36 1

Numbers without a way of being formed, such as 13 or 14, are not shown.
So the best 6 numbers to have on your card are 1,2,3,4,5 and 6.
The different dice falls that produce these first 6 numbers are:
1x 1 1+ 1 1+ 2 1+ 3 1+ 4 1+ 5
1 2 1x 2 1x 3 1x 4 1x 5 1x 6
2 1 1 3 1 4 1 5 1 6 2x 3
2/ 2 2x 1 2+ 1 2x 2 2+ 3 2+ 4
2 3 2 4 2 5 2 6 3+ 2 3x 2
3 2 3 1 2/ 6 3+ 1 4+ 1 3+ 3
3/ 3 3 5 3x 1 4x 1 5x 1 4+ 2
3 4 3/ 6 3 6 5 1 6 1 5+ 1
4 3 4 2 4 1 6 2 6x 1
4/ 4 4 6 5 2
4 5 5 3 6/ 2
5 4 6/ 3 6 3
5/ 5 6 4
5 6
6 5
6/ 6

The first column shows falls that produce 1; the second,2; etc.
Note that 2,1 is different from 1,2 as they are different ways that the dice can fall. The indication of the operator between them is not meant to imply the order in which the numbers are given to the operation. That is, for example, 2/6 and 6/2 are both shown for producing 3, as they are different falls of the dice that can produce 3 by division, but obviously 2 has to be the divisor and 6 the dividend.
Also note what is important: a given fall of the dice can be counted only once. For example, a fall of 2,2 counts for just one way of producing 4; you can make it addition or make it multiplication, but it still counts as only 1 way, as we're counting ways of the dice falling, not your choice of how to make the number. Only one way is shown on the above chart, to reflect that fact.

Posted by Charlie
on 20030722 12:40:52 