A nice dice problem-Part 2

In my last post (see here) I posed a dice problem, promising to give the answer in the next blog which is this blog. Here is the problem from my last blog:

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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 in the comments 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.

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Some pips on this:

a) This problem was posed and solved in a Martin Gardner column, and later generalized by Gallian and Rusin (the paper is here). I did a write up of just the two 6-sided dice case and a few other things here. I also have slides on the problem here.

b) The only answer is (1,2,2,3,3,4) and (1,3,4,5,6,8).  Here is the first step of the solution, though better off reading my writeup pointed to in item a.

We seek \(a_1,\ldots,a_6\) and \(b_1,\ldots,b_6\)  (NOT (1,2,3,4,5,6)) such that

\((x^{a_1} + \cdots + x^{a_6})(x^{b_1}+ \cdots + x^{b_6}) = (x^1 + \cdots + x^6)^2 \).

Using polynomials also lets you solve the problem for other types of dice, as shown in my writeup, my slides, and the original paper. (Spellcheck thinks writeup is not a word. I disagree.)

c) How did I find this question, and its answer, at random? I intentionally went to the math library, turned my cell phone off, and browsed some back issues of the journal Discrete Mathematics. I would read the table of contents and decide what article sounded interesting, read enough to see if I really wanted to read that article. I then SAT DOWN AND READ THE ARTICLES, taking some notes on them. 

d) This is more evidence that perhaps we should unplug, at least partially, sometimes. I blogged about that here.

e) Technology is NOT the real issue here. Its allowing yourself the freedom to NOT work on a a paper for the next STOC/FOCS/WHATNOT conference and just read math for FUN without thinking in terms of writing a paper. I am supposed to say you never know when random knowledge may help you get a result but that's the wrong mentality since it circles back to being non-random as you will only read things that you think will lead to results.

f) The STEM library at UMCP stopped getting PAPER journals a while back. Hence the only journals I can look at are before a certain date. Should the STEM library subscribe to paper journals just so I can read journals at random? OF COURSE NOT. Instead I may in the future surf the web in an intelligent way looking for random article I find interesting. I have sometimes used arxivs for this, though I also sometimes go into a NON-PRODUCTIVE black hole.

g) Older journals are sometimes more readable. The STEM library still has plenty of them, so I might not need to use the web for this for a while. 

h) For the purposes of random browsing, journals that are focused and whose name says what their focus is (e.g., Discrete Math, Journal of Symbolic Logic, Conference on Topological Algebraic Topology) are good since you know what's in them and hence if you care (or have enough background knowledge). By contrast, a journal title like Proceedings of he London Math Society or Duke Mathematics Journal or Southern North Dakota Math Journal don't tell you anything about what's in them, so its not worth looking at for this purpose.

i) SO, is the dice problem I posted on well known. From the comments on the original post, and my own observations, here are arguments for YES and for NO.

YES, Well known: (i) There is a name for these kind of dice, Sicherman dice. (ii) There is a Wikipedia page on Sickerman dice here. (iii) You can buy Sicherman dice here. (iv) The problem is in the book Concrete Mathematics as an exercise in Chapter 8, page 431. (v) There a Martin Gardner column, so it was well known at one time (though that may have faded). 

NO, Not Well known: (i) I had not heard of it and I've been around math and dice (I have a paper on loaded dice) for a long time. (ii) Lance had not heard of it. (iii) Some of my readers had not heard of it.

UPSHOT: The notion of well known is not well defined; however, if some of my readers did not know it, and are now enlightened, then I am happy.