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Making equations with six six-sided dice

  1. Jun 3, 2012 #1
    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:
    Another one is:

    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. jcsd
  3. Jun 3, 2012 #2
    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?
  4. Jun 4, 2012 #3
    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.
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