Making equations with six six-sided dice

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In summary, the conversation discusses a game where a player must use combinations of basic operations to create equations with numbers rolled on dice. It is mentioned that not all combinations can be made into valid equations for up to five dice, and the question is posed about whether there are any combinations that cannot be made into valid equations for six dice. The minimum number of dice needed to guarantee a valid equation is also discussed. The conversation concludes with the thought that it would be useful to consider each operation separately and to use results from number theory.
  • #1
Mingy Jongo
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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?
 
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  • #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?
 
  • #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.
 

1. How do I make an equation using six six-sided dice?

To make an equation using six six-sided dice, you will need to roll the dice and assign a mathematical operation to each number rolled. For example, if you roll 1, 2, 3, 4, 5, and 6, you could assign addition, subtraction, multiplication, division, exponentiation, and square root to each number respectively. This would result in an equation like (1+2-3*4/5)^6 = ?

2. Can I use all six dice in one equation?

Yes, you can use all six dice in one equation. This is what makes this activity challenging and fun, as you have to think creatively to use all the numbers rolled in a single equation.

3. Are there any specific rules for making equations with six six-sided dice?

There are no specific rules for making equations with six six-sided dice. The only requirement is that you use all six dice in one equation and assign a mathematical operation to each number rolled.

4. Can I use any mathematical operation in my equation?

Yes, you can use any mathematical operation in your equation, such as addition, subtraction, multiplication, division, exponentiation, square root, etc. However, it is recommended to use a variety of operations to make the equation more challenging and interesting.

5. Is there a correct or incorrect answer for making equations with six six-sided dice?

There is no correct or incorrect answer for making equations with six six-sided dice. The goal of this activity is to have fun and challenge your mathematical thinking skills. As long as you use all six dice in one equation and follow mathematical rules, any answer is acceptable.

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