# Making equations with six six-sided dice

1. Jun 3, 2012

### Mingy Jongo

Suppose I am playing a game where I roll some dice, and must use only combinations of elementary operations (addition, subtraction, multiplication, or division) to make an equation using each number rolled exactly once.

For example, if I roll four six-sided dice, and get this:
2, 3, 3, 4
A possible solution is:
3-2=4-3
Another one is:
(3+3)-2=4

Now, I'm fairly certain that for up to five six-sided dice, not all combinations can be made into valid equations. Here are some examples:
2 dice: Everything except doubles
3 dice: Triples greater than 1s; two 1s and a 3, 4, 5, or 6
4 dice: Three 1s and a 4, 5, or 6
5 dice: Four 1s and a 5 or 6

My question is, are there any combinations that cannot be made into valid equations for six six-sided dice? Five ones and a six works here: (1+1)*(1+1+1)=6
If so, can someone give me an example of such a combination, and more interestingly, the minimum number of six-sided dice needed to guarantee a valid equation for any possible roll, if there indeed exists a minimum?

2. Jun 3, 2012

### Matt Benesi

Consider that for any number of rolls above 6, you will always have a duplicate die which can be counted as 1 (via division of the duplicate by itself).

Definitely a neat thing to think about, wonder if it or something similar has come up here before?

3. Jun 4, 2012

### Number Nine

What a wonderful problem; it will give me something to do over the summer.
It seems to me that it would be easiest/most informative to consider each operation separately; for example, when considering addition, we're really asking how may roles generate exactly two partitions of the same number, which opens the door to using results from number theory.