## Sunday, January 07, 2024

### A nice dice problem- Part I

In this blog I pose a dice problem. The problem is NOT mine and the answer is KNOWN. However, I DO NOT  think its well known, and I DO think it's interesting. My next post will have the answer.

PROBLEM:

A 6-sided fair die is a 6-tuple of positive natural numbers. (NOTE- POSITIVE NATURAL NUMBERS SO YOU CANNOT USE ZERO. I am emphasizing this since some answers given used 0.)

The standard 6-sided die is (1,2,3,4,5,6).

Do there exist two 6-sided dice  $$a_1,\ldots,a_6$$ and $$b_1,\ldots,b_6$$  (the numbers need not be distinct) such that

a) The dice  are NOT standard

b) When you roll the two dice you get THE SAME probabilities of sums as rolling two standard dice (we are assuming the dice are fair so the prob of any side is 1/6, though if a die has two faces with 4 pips on them, then of course the prob will of getting a 4 will be 1/3). So, for example, the probability of getting a sum of 2 is 1/12, the probability of getting a sum of 7 is 1/6.

A few pips for now:

a) Try to get a method to solve it for two d-sided dice. Or other generalization.

b) You may post the answer or whatever thoughts on the problem you want; however, if you want to solve it without any hints, don't look at the comments.

c) I do not know how hard this problem is to solve since I read the solution before really trying to solve it. I do not know how hard it is to find the answer online since I found the paper at random.

d) I think the problem is not well known. I could be wrong. Leave comments if you've heard of it or not heard of it or whatever, to comment on that point. Note that well known is not a well defined term.

1. Die 1: 0,1,2,3,4,5
Die 2: 2,3,4,5,6,7

2. Unless I have made a mistake (very possible/not unlikely), this is impossible: this corresponds to factoring the polynomial $x^2+2x^3+3x^4+4x^5+5x^6+6x^7+5x^8+4x^9+3x^{10}+2x^{11}+x^{12}$ as the product $p(x)q(x)$ of two degree-6 polynomials with no constant term and positive coefficients. And there is only one factorization of this type...

1. (Bill) Why that polynomial? You are correct that the answer involves polynomials. But the problem is asking if there is a_1,...,a_6,b_1,...,b_6 such that (x^{a_1} + ... + x^{a_6})(x^{b_1}+...+x^{b_6}) is equal to (x^1 + ... + x^6)^2 but not in the obvious way.

2. You're right -- I was solving the wrong problem. (I misread the question as "is there a set of non-uniform probabilities for the usual labels" instead of "is there a set of non-standard labels with uniform probabilities")

3. (Bill) The labels on the dice have to be POSITIVE naturals, so you have not solved the problem as stated.

4. Thanks for the link! I didn't know it was there since I didn't know they were called Sicherman dice. I think having a Wikipedia link is a necc but not suff condition for a problem to be well known. So is this dice problem well known? The Jury is still out. Lance didn't know it, so thats evidence for no. (OH- This is bill.)

5. Evangelos Georgiadis9:43 PM, January 07, 2024

(a) A hint is to think about how irreducible factors of the polynomial
(z^1+z^2+z^3+z^4+z^5+z^6)^n with rational coefficients might play a role in this.

(b) A solution is (1,3,4,5,6,8) and (1,2,2,3,3,4).

(c) In terms of hardness of problems, our good friends, Don Knuth, Oren Patashnik and the late Ron Graham
considered it to be an exam type problem -- serving nicely as a take home exam (not for in-class assessments under time pressure).
In terms of source; (not sure about the paper mentioned but it appeared in GKP (Concrete Mathematics) as exam problem 36 in Chapter 8, on page 431. The way it has been phrased in this post, might be slightly more challenging because no initial configuration was provided.

6. Z3 found a solution in <1 second. https://ideone.com/oXXw7C

Die 1
1 2 2 3 3 4
Die 2
1 3 4 5 6 8

7. I'd say well known since you can buy a pair of Sicherman dice, say at https://www.mathartfun.com/dSpecial.html.