Transcendental Dice Rolls

Three fair 6-sided dice each have their sides labeled0,1,*e*,*π*,*i*, (2)^(1/2). If these dice are rolled, the probability that the product of all the numbers on the top face is real can be expressed as *a**b*, where *a*and *b* are coprime positive integers. What is the value of *a*+*b*?

I am not able to think much more over it as what to these SPECIAL numbers....

Re: Transcendental Dice Rolls

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Originally Posted by

**geniusgarvil** Three fair 6-sided dice each have their sides labeled 0,1,e,π,i, (2)^(1/2). If these dice are rolled, the probability that the product of all the numbers on the top face is real can be expressed as ab, where aand b are coprime positive integers. What is the value of a+b?

The difficulty here is the word *co-prime*.

Not everyone accepts this convention, "The numbers 1 and −1 are coprime to every integer, and they are the only integers to be coprime with 0." See this page.

Thus any triple that contains at least one zero has the given property.

There are $\displaystyle 6^3-5^3$ of those.

But so does the triple $\displaystyle (1,1,1)$, as does $\displaystyle (1,\sqrt2,\sqrt2)$ (there are three of those).

The triple $\displaystyle (1,i,i)$ also (three of them).

Now I have no idea why it would ask "What is the value of a+b?"

For many reasons, I think this is a flawed question.

Re: Transcendental Dice Rolls

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**Plato** The difficulty here is the word *co-prime*.

I am not sure what the difficulty is. The property in question is not that the numbers on the top face are coprime, but that the product of those numbers is real. This happens iff i appears exactly 0 or 2 times.

My guess the reason the question asks for a + b where a / b is the required probability is that there is an automated system that accepts one integer number only, so it is not possible to enter a/ b directly. Of course, a / b is the actual answer.

Re: Transcendental Dice Rolls

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**emakarov** I am not sure what the difficulty is. The property in question is not that the numbers on the top face are coprime, but that the product of those numbers is real. This happens iff i appears exactly 0 or 2 times.

I understand about the product. The product must equal the product of integers that are co-prime.

The difficulty is that I have seen an author who does not think that zero is co-prime with any integer. I can also think that may be true of one.

But I do accept the Wikipedia convention.

Also $\displaystyle e\text{ or }\pi$ cannot appear at all.

We could have $\displaystyle (1,\sqrt2,\sqrt2)$, but not $\displaystyle (\sqrt2,\sqrt2,\sqrt2)$.

Re: Transcendental Dice Rolls

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Originally Posted by

**geniusgarvil** If these dice are rolled, the probability that the product of all the numbers on the top face is real can be expressed as ab, where a and b are coprime positive integers.

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Originally Posted by

**Plato** I understand about the product. The product must equal the product of integers that are co-prime.

No, the question doesn't ask to find the probability that the product of the top numbers is a product of coprime integers. Rather, it asks to find the probability that the product of the top numbers is *real*, and that probability is expressed as a / b where a and b are coprime.

Only after writing this I noted that the OP wrote ab instead of a / b; hence the confusion. I still think that my interpretation is right. Similar questions about finding a fraction and requesting the sum of the numerator and the denominator have been posted before. I know what you are thinking. Just please don't kill the OP. (Smile)

Re: Transcendental Dice Rolls

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Originally Posted by

**emakarov** No, the question doesn't ask to find the probability that the product of the top numbers is a product of coprime integers. Rather, it asks to find the probability that the product of the top numbers is *real*, and that probability is expressed as a / b where a and b are coprime.

Only after writing this I noted that the OP wrote ab instead of a / b; hence the confusion. I still think that my interpretation is right. Similar questions about finding a fraction and requesting the sum of the numerator and the denominator have been posted before. I know what you are thinking. Just please don't kill the OP. (Smile)

I am really against guessing at what the OP really meant. That is not what is posted, is it?

Re: Transcendental Dice Rolls

I agree about not guessing. Usually by responding to the straightforward but probably wrong interpretation of the question (e.g., when necessary parentheses were omitted), I hope to shame the OP and make him/her work harder. I just want to point out that the phrase "the probability that the product of all the numbers on the top face is real can be expressed as ab" does not mean that the product can be expressed as ab: in this case it has to say say, "the probability that the product... is real *and* can be expressed...". So it is the probability that can be expressed, not the product of the top numbers.

Re: Transcendental Dice Rolls

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Originally Posted by

**emakarov** I agree about not guessing. Usually by responding to the straightforward but probably wrong interpretation of the question (e.g., when necessary parentheses were omitted), I hope to shame the OP and make him/her work harder. I just want to point out that the phrase "the probability that the product of all the numbers on the top face is real can be expressed as ab" does not mean that the product can be expressed as ab: in this case it has to say say, "the probability that the product... is real *and* can be expressed...". So it is the probability that can be expressed, not the product of the top numbers.

I just disagree.