# Four dice puzzle: What's the best throw?

This continues Damiano's puzzle "Four dice puzzle: 2,2,4,5"

Damiano keeps throwing his four dice. After a lot of throwing and thinking and working, he has determined for every throw $a,b,c,d$ of his four dice the smallest positive integer $N(a,b,c,d)$ that cannot be generated from this throw according to the following rules:

• One may use the four numbers $a,b,c,d$ in any order, and it is fine if not all of them are used.
• Concatenation of digits is NOT allowed.
• The only allowed operations are additions, subtraction, multiplication, and division ($+,-,*,/$).
• One may use any number of brackets.

Question: Which throw $a,b,c,d$ of dice yields the largest number $N(a,b,c,d)$ ?

2, 4, 5, 6

Which yields an N of

45

Barring a bug in my test program, this should be correct. I tried all possible values.

Here are all the values for my answer:

1 = 5 - 4
2 = 2
3 = 5 - 2
4 = 4
5 = 5
6 = 6
7 = 5 + 2
8 = 6 + 2
9 = 5 + 4
10 = 6 + 4
11 = 6 + 5
12 = 6 * 2
13 = 6 + 5 + 2
14 = (5 * 4) - 6
15 = 6 + 5 + 4
16 = (6 * 2) + 4
17 = (6 * 2) + 5
18 = (5 + 4) * 2
19 = (6 * 4) - 5
20 = 5 * 4
21 = ((6 * 4) + 2) - 5
22 = (5 * 4) + 2
23 = (5 * 4) + (6 / 2)
24 = 6 * 4
25 = ((6 + 4) * 2) + 5
26 = (6 * 4) + 2
27 = ((6 * 4) + 5) - 2
28 = (5 + 2) * 4
29 = (6 * 4) + 5
30 = 6 * 5
31 = (6 * 4) + 5 + 2
32 = (6 * 5) + 2
33 = ((2 / 4) + 5) * 6
34 = (6 * 5) + 4
35 = ((6 / 2) + 4) * 5
36 = (4 + 2) * 6
37 = ((6 + 2) * 4) + 5
38 = ((5 + 2) * 6) - 4
39 = ((5 / 2) + 4) * 6
40 = (6 + 2) * 5
41 = ((4 + 2) * 6) + 5
42 = (5 + 2) * 6
43 = (6 * 4 * 2) - 5
44 = (6 + 5) * 4
45 = ???

• 45 = (6*6)+5+4 or 45 = (6+3) * 5 or am I missing something? – VenomFangs Oct 12 '15 at 19:16
• @VenomFangs You are missing something. Neither of your 2 options can be done with the 4 numbers I used to get 1-44. For example, 6,6,5,4 can get you 45, but it won't get you 33, so it isn't the best option. – Joel Rondeau Oct 12 '15 at 19:20
• Gotcha, the question was more in depth than I was giving it credit for. Thanks! – VenomFangs Oct 12 '15 at 19:23