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Dicey Permutations


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#1 superprismatic

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Posted 23 October 2012 - 12:15 AM

If we have any 4-tuple of distinct real
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?
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#2 CaptainEd

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Posted 23 October 2012 - 09:08 PM

What an interesting puzzle! Unfortunately, I don't see how to make incremental progress on it.
Spoiler for What I've learned so far

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#3 phil1882

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Posted 23 October 2012 - 10:32 PM

Spoiler for would this work?

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#4 CaptainEd

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Posted 23 October 2012 - 10:47 PM

Spoiler for D1 needs to be lowest 1/4 of the time

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#5 superprismatic

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Posted 24 October 2012 - 03:09 PM

Spoiler for would this work?

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.
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#6 EventHorizon

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Posted 01 November 2012 - 12:46 AM

Spoiler for initial thoughts


Spoiler for 3 dice fewest faces


Spoiler for 4 dice progress

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#7 superprismatic

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Posted 01 November 2012 - 08:30 PM

Spoiler for initial thoughts


Spoiler for 3 dice fewest faces


Spoiler for 4 dice progress

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.
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#8 EventHorizon

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Posted 01 November 2012 - 09:50 PM

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

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#9 superprismatic

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Posted 02 November 2012 - 12:07 AM

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

Nice! I just checked your 30-face solution and it works just fine. I hope you can go lower.
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#10 superprismatic

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Posted 04 November 2012 - 12:38 AM

Spoiler for my report

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