# Each of the six faces of a die is marked with an integer, not necessarily positive.?

Each of the six faces of a die is marked with an integer, not necessarily positive. The

die is rolled 1000 times. Show that there is a time interval such that the product of all rolls

in this interval is a cube of an integer. (

### 3 Answers

- roderick_youngLv 71 year ago
Might be possible. Let me assume that the faces are a, b, c, d, e, and f.

We cannot roll 3 of the same face in a row, or else that would be the perfect cube. So no a a a.

Now how about sequences of the form, abc abc. There are 6^3 possible triples, so 216. But order doesn't matter to me in this case, so abc is as good as bac. So divide by 3!, to get 30 possible triples. But I cant use aaa, bbb, etc., so subtract 6 to get 24 possible triples.

I consider each triple now to be a letter. I have a string of 24 letters. I want to make it so that no two adjacent letters are the same. Hold on, let me think about this.

have u thought of the ending yet ? I agree we denote the numbers as letter a-f we cant have a string like abcdedfcabdefgabcabcabcdefg because one big jump out at you cube in this string is the abcabcabc. pardon my g's