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# Dicey Permutations

### #1

Posted 23 October 2012 - 12:15 AM

numbers, there is a simple way to have

this determine a permutation on 4 things:

Just replace each number with its ranking

amongst the 4 numbers. For example,

suppose I had the 4-tuple, (36,95,1,18).

By replacing each number with its rank,

I get the permutation (2,1,4,3).

I would like to be able to use 4 fair

dice to generate a permutation in this

way. The dice would give me the 4-tuple

(each die would have its own spot in the

tuple) and I would use the ranks to

determine a permutation. Of course, In

order to insure that I get 4 distinct

numbers, no die can have a number which

is on any other die. But it may be the

case that a paricular die has two or more

faces having the same value. The number

of faces on the dice may be any positive

integer (I'm assuming that fair dice can

always be made this way). It is not

necessary that all of the dice have the

same number of faces.

Can you construct a set of 4 dice which

can produce all 24 permutations of 4

things, each with probability 1/24 ?

I have several such sets with 12 faces

on each die. Can you find a set with

fewer total faces?

### #2

Posted 23 October 2012 - 09:08 PM

### #3

Posted 23 October 2012 - 10:32 PM

### #4

Posted 23 October 2012 - 10:47 PM

### #5

Posted 24 October 2012 - 03:09 PM

No, that can't work because there are 24 permutations of 4 things and your dice can produce 256 different results, but 256 results can't be split evenly amongst 24 permutations. Some permutations would have to get more dice results than others.Spoiler for would this work?

### #6

Posted 01 November 2012 - 12:46 AM

### #7

Posted 01 November 2012 - 08:30 PM

Nice solution for 3 dice! I hope you can beat my 4 dice solution of 48 faces. I suspect a solution with fewer than 48 faces exists, but I haven't found one.Spoiler for initial thoughtsSpoiler for 3 dice fewest facesSpoiler for 4 dice progress

### #8

Posted 01 November 2012 - 09:50 PM

### #9

Posted 02 November 2012 - 12:07 AM

Nice! I just checked your 30-face solution and it works just fine. I hope you can go lower.So, after reading your post and looking at the fewest for 3 again. I thought of one configuration to try. Since testing a configuration with my code is fast, I threw it into my code and...

Spoiler for perhaps not the least, but better than 48

### #10

Posted 04 November 2012 - 12:38 AM

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