# Four dice puzzle: Postscriptum

This continues (and completes) the sequence of four dice puzzles:
Four dice puzzle: 2,2,4,5
Four dice puzzle: What's the best throw?

The days go by and Damiano is still throwing his four dice. He has determined for every throw $a,b,c,d$ of his four dice (with $a\le b\le c\le d$) 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.

Questions:

1. Which throw $a,b,c,d$ of dice yields the smallest number $N(a,b,c,d)$ ?
2. How many different throws $a,b,c,d$ have $N(a,b,c,d)\ge40$ ?
3. Which throws $a,b,c,d$ have $N(a,b,c,d)=28$ ?
4. What is $N(6,6,6,6)$ ?
• The simplest way to answer all four questions would be to just write a program that computes all possible $N(a,b,c,d)$ – ghosts_in_the_code Oct 14 '15 at 10:04

I wrote a program that calculates all N(a,b,c,d). This is my result:

N(1,1,1,1) = 5
N(1,1,1,2) = 7
N(1,1,1,3) = 10
N(1,1,1,4) = 11
N(1,1,1,5) = 13
N(1,1,1,6) = 15
N(1,1,2,2) = 10
N(1,1,2,3) = 11
N(1,1,2,4) = 14
N(1,1,2,5) = 17
N(1,1,2,6) = 20
N(1,1,3,3) = 14
N(1,1,3,4) = 19
N(1,1,3,5) = 22
N(1,1,3,6) = 26
N(1,1,4,4) = 22
N(1,1,4,5) = 27
N(1,1,4,6) = 13
N(1,1,5,5) = 13
N(1,1,5,6) = 14
N(1,1,6,6) = 15
N(1,2,2,2) = 11
N(1,2,2,3) = 17
N(1,2,2,4) = 19
N(1,2,2,5) = 23
N(1,2,2,6) = 19
N(1,2,3,3) = 22
N(1,2,3,4) = 29
N(1,2,3,5) = 19
N(1,2,3,6) = 28
N(1,2,4,4) = 21
N(1,2,4,5) = 33
N(1,2,4,6) = 29
N(1,2,5,5) = 29
N(1,2,5,6) = 44
N(1,2,6,6) = 27
N(1,3,3,3) = 14
N(1,3,3,4) = 17
N(1,3,3,5) = 22
N(1,3,3,6) = 23
N(1,3,4,4) = 22
N(1,3,4,5) = 29
N(1,3,4,6) = 34
N(1,3,5,5) = 34
N(1,3,5,6) = 37
N(1,3,6,6) = 20
N(1,4,4,4) = 10
N(1,4,4,5) = 18
N(1,4,4,6) = 31
N(1,4,5,5) = 12
N(1,4,5,6) = 17
N(1,4,6,6) = 21
N(1,5,5,5) = 8
N(1,5,5,6) = 13
N(1,5,6,6) = 14
N(1,6,6,6) = 9
N(2,2,2,2) = 7
N(2,2,2,3) = 13
N(2,2,2,4) = 11
N(2,2,2,5) = 17
N(2,2,2,6) = 15
N(2,2,3,3) = 19
N(2,2,3,4) = 17
N(2,2,3,5) = 29
N(2,2,3,6) = 23
N(2,2,4,4) = 11
N(2,2,4,5) = 23
N(2,2,4,6) = 17
N(2,2,5,5) = 28
N(2,2,5,6) = 23
N(2,2,6,6) = 21
N(2,3,3,3) = 13
N(2,3,3,4) = 25
N(2,3,3,5) = 29
N(2,3,3,6) = 26
N(2,3,4,4) = 23
N(2,3,4,5) = 41
N(2,3,4,6) = 31
N(2,3,5,5) = 29
N(2,3,5,6) = 38
N(2,3,6,6) = 23
N(2,4,4,4) = 11
N(2,4,4,5) = 25
N(2,4,4,6) = 15
N(2,4,5,5) = 26
N(2,4,5,6) = 45
N(2,4,6,6) = 17
N(2,5,5,5) = 14
N(2,5,5,6) = 24
N(2,5,6,6) = 25
N(2,6,6,6) = 17
N(3,3,3,3) = 11
N(3,3,3,4) = 17
N(3,3,3,5) = 19
N(3,3,3,6) = 13
N(3,3,4,4) = 18
N(3,3,4,5) = 37
N(3,3,4,6) = 29
N(3,3,5,5) = 32
N(3,3,5,6) = 19
N(3,3,6,6) = 14
N(3,4,4,4) = 10
N(3,4,4,5) = 22
N(3,4,4,6) = 29
N(3,4,5,5) = 31
N(3,4,5,6) = 43
N(3,4,6,6) = 17
N(3,5,5,5) = 11
N(3,5,5,6) = 29
N(3,5,6,6) = 25
N(3,6,6,6) = 11
N(4,4,4,4) = 10
N(4,4,4,5) = 14
N(4,4,4,6) = 13
N(4,4,5,5) = 22
N(4,4,5,6) = 32
N(4,4,6,6) = 19
N(4,5,5,5) = 7
N(4,5,5,6) = 13
N(4,5,6,6) = 33
N(4,6,6,6) = 13
N(5,5,5,5) = 8
N(5,5,5,6) = 13
N(5,5,6,6) = 3
N(5,6,6,6) = 8
N(6,6,6,6) = 9


And now to answer the questions, assuming my program made no mistakes:

Which throw a,b,c,d of dice yields the smallest number N(a,b,c,d) ?

N(5,5,6,6) = 3

How many different throws a,b,c,d have N(a,b,c,d)≥40 ?

four:
N(1,2,5,6) = 44
N(2,3,4,5) = 41
N(2,4,5,6) = 45
N(3,4,5,6) = 43

Which throws a,b,c,d have N(a,b,c,d)=28 ?

only (1,2,3,6) and (2,2,5,5)

What is N(6,6,6,6) ?

9

• By the way. my code (Lua) can be seen here pastebin.com/311G0YK9 . It's not that good I guess and slow but it works – Ivo Beckers Oct 14 '15 at 12:31