Download Prior work in problems related to dice

Prior work in problems related to dice
Prior work on the total-to-parts map has all focussed on either the standard
or the fair fiber. The main reference [15] for the former is reviewed in our paper.
The fair fiber has a longer history, dating back to 1951, and has sparked more
work. We review this literature in the first subsection. Although [15] answers the
title question of our paper, the game of craps is never mentioned in it. We have,
however, found some prior work dealing with craps that is reviewed in the second
subsection. Finally, the last subsection discusses work on sacks of fair dice whose
sides have non-standard labelings but whose total distribution is standard; this falls
outside the class of problems we study, but it includes the question of Stoyanov
that sparked our work. We use without comment notation from our paper.
Strict sacks in the fair fiber We digress briefly to discuss the very first result,
from 1951, on fibers of the total-to-parts map that we have been able to locate, and
some generalizations. We call the fiber lying over the distribution with all totals
equally likely the fair fiber. In this case we have f( x ) = T1 ψT ( x ) whose one real
root is, for odd T only, −1. A factorization of this polynomial will have all factors
real if and only if each factor d( x ) is itself a product of certain of the χm,T ( x ) and,
necessarily, of x + 1, if d( x ) has even order k. Hence,
Proposition 1. The fair fiber of fk contains a real sack if and only if T is odd and exactly
one k j is even or T is even and no k j is even.
Monthly problem E 925 [13], posed by J. B. Kelly asked, in our language, whether
the fair fiber for a pair of cubical dice contains any strict sacks. The negative solution
by J. V. Finch and P. R. Halmos is essentially the argument above. The second
approach cited, due Leo Moser and J. H. Wahab, uses the assumed equality of the
lowest and highest totals to obtain a contradiction. Both arguments are repeated
without attribution in the solution of Problem 7A of [12].
Dudewicz and Dann [7], although they do not cite [13], note that, for i.i.d. cubical
dice, the conclusion is a well-known exercise. They cite the text of Parzen [14],
where this is Problem 9.12. Their title suggests that there are no strict sacks in
any fair fiber, but they prove this only for sacks of n dice of equal order k by
1
showing that f k−1 > T +
1 . Their result is reproved (but not cited) by Chen, Rao and
Shreve [5] by showing that there must be a pair (t, t0 ) with | f t − f t0 | ≥ nn−2 k1 .
The last word on this question is provided by Gasarch and Kriskal [11] who
define a strict die (or sack) to be nice if its distribution is palindromic and any
positive probability equals p0 (or f 0 ); for example, a fair sack is nice. They show
that a strict sack is fair if and only if each die d is nice and each total is unique (arises
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Prior work in problems related to dice
from exactly 1 roll with non-zero probability). The sufficiency of these conditions is
clear. The necessity of palindromicity follows from the fact that inversion preserves
the roots of each χm,T ( x ) and of x + 1, and hence also any real d( x ) dividing ψT ( x ).
The key step (Lemma 4) is that a nice pair consists of 2 nice dice; an easy induction
(Theorem 4), then shows that a nice sack contains only nice dice, although this is
stated only for fair sacks. All rolls with non-zero probability of a sack of nice dice
share the same probability, and a total of 0 arises exactly by rolling 0 on all dice, so
if the sack is fair, then each total must be unique. Since a nice die is determined by
the subset of sides having positive probability, there are only finite many nice dice
of each order and hence only finitely many sacks of nice dice of any combinatorial
type. This yields an effective method of finding all strict sacks in any fair fiber.
Question 2. Can one describe explicitly the set of combinatorial types k whose fair fibers
contain strict sacks?
Papers that discuss craps The earliest paper we have found dealing with craps is
one from 1919 by Bancroft Brown [1]. This begins with a table showing, without
any derivation, the expected number of occurrences of a win or loss from each
initial roll in 9900 = 495 · 20 games. The reader who has reviewed our discussion [6,
1.craps], reviewing the rules of craps and calculating the probability of winning
will recognize the number 495 as the denominator of that probability. An additional
factor 4 is needed so that all these numbers in the table will be integral, but the
additional factor of 5 is superfluous. The table also provides the empirical number
of occurrences observed in 9900 trials of the game. Brown then derives the expected
number of rolls in a game as in the exercise at the end of [6, 1.craps] and concludes
by computing the expected number of passes, defined as games up until the first
loss by failing to make a point (but not by throwing a crap) and comparing the
distribution of the number of passes with its empirical approximation based on a
sample of 1000 passes.
Example 3. Show that the expected number of passes is
exercise at the end of [6, 1.craps] applies.
495
196 .
Hint: Part (1) of the
The paper [17] of Armand Smith begins by computing the probability of winning
at craps by summing the geometric series that arise from the tree-diagram (and its
analogues)in [6, 1.craps]. By grouping the terms in these series corresponding to
each depth in the tree, he then finds the distribution of the number of rolls and uses
it to compute both its mean and variance by explicit series computations.
Finally, the paper of Bryson [4] considers the effect of varying the probability
p of rolling a total of 7, while keeping the relative probabilities of the other totals
unchanged, but without worrying about whether dice that realize this total distribution exist. Indeed, our reviewers called to our attention a cryptic assertion
attributed by Bryson to his referee asserting that no realization of such total by a
6
pair of dice is possible, except, for p = 36
, by using a pair of fair dice.
Prior work in problems related to dice
3
This is indeed so, for generic perturbations, if the dice are required to be identical
6
but not if they may differ. Write p = 36
+ 30e giving a perturbed fe ( x ) of
4
fe ( x ) =
1
6
∑ (i + 1)( 36 − e)(xi + x10−i ) + ( 36 + 30e)x5
i =0
and note that this is both real (hence complex roots occur in conjugate pairs)
and palindromic (hence all roots occur in pairs that are multiplicative inverses).
The former assertion may then be checked either by writing down the system
of equations in e and the side probabilities pi that a pair of identical dice would
have to satisfy and eliminating the pi to see that e must be 0 or by computing the
discriminant of fe ( x ) and verifying that it is not identically 0 as a function of e. If so,
then fe ( x ) cannot be a square, and hence no identical dice give these totals. To see
that nonidentical dice can have totals fe ( x ), it suffices to check that the fe (−1) < 0
and hence that the pair of roots −1 of f0 ( x ) deform to a pair of mutually inverse real
roots of fe ( x ), one less than and one greater than −1, and not to a pair of complex
conjugate roots. By collecting one of these roots with two of the complex conjugate
pairs of roots of fe ( x ) and the other with the remaining two pairs, we produce a
factorization of fe (−1) < 0 into a sack of two real dice. Since all coefficients of f0 ( x )
are positive, these factorizations are strict for small e.
Bryson finds that reducing p raises the chance of winning, while increasing p
first lowers this chance until p reaches roughly 0.23, then raises it; also, the player’s
chance of winning is less than 12 only for p in the approximate interval (0.13, 0.37).
He also shows that the effect of making raising the probabilities the sides with large
numbers (either 4-6 or 3-6) on one or both dice and lowering the probabilities of
the other side is to raise the players probability of winning, and vice-versa.
Papers on Sicherman dice The question of when there exist sacks of fair dice
whose sides have non-standard labelings (with a positive integral number of spots
on each side) but whose total distribution is standard, seems to have first been posed
and solved for pairs and triples of six-sided dice by George Sicherman and was
popularized in—was there ever any doubt!—a 1978 Martin Gardner Mathematical
Games column [10, p.19]. Sicherman showed that there is, up to permutation, a
unique non-standard pair with sides marked (1, 2, 2, 3, 3, 4) and (1, 3, 4, 5, 6, 8) and
that no non-standard triple exists. Although it is now standard to refer to such dice
(and generalizations) as Sicherman dice, his example seems to have been forgotten,
and rediscovered, many times. The example is given in the books of Stoyanov
[18, 12.7] and Halmos [12, Problem 7C] and in the paper of Robertson, Shortt and
Landry [15, p.319] but none of these mention Sicherman or Gardner1.
1The example is attributed to W. W. Funkenbusch in [15], and given with no citation in [12] and [18].
The latter book, which terminates with over 20 pages of references, contains no citations, but as the
paper [15] is among these references, we suspect that Stoyanov learned of the example from it. Had this
citation been given, we would have been led to [15] at the start, rather than the end, of our work.
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Prior work in problems related to dice
The example prompted two sequels the following year, both dealing with Sicherman sacks S of n dice of equal order k (though neither uses these letters to denote
these quantities). Both note that, by (2), the polynomial d( x ) of any die in such a
n
set must divide the standard total polynomial which is ψk ( x ) . Broline [2] then
provides tables of the possible side numberings when n is the number of sides of
a Platonic solid, a restriction which has nothing to do with the arguments but is
motivated by the desire to use such solids as physical models.
The model of Gallian and Rusin [9] is the generalized dreidel shown on the
cover of Mathematics Magazine, Vol. 49, No. 3 so they work with general n. Their
paper covers too many special cases to list fully here, but one main thrust is to
list all dice that can appear in such an S when k has a simple prime factorization.
In this direction, they show: that only the standard die is possible if k is prime
(Theorem 3); that there are exactly 3 possible dice if k has, with multiplicities, at
most 2 prime factors (Theorem 2), which they write down if one factor is 2 or both
`−1
`
are equal (Corollaries 1 and 2); and that there are (2`−
1 ) possible dice of order p
(Theorem 10). They then prove various equalities and inequalities relating the
labels of a Sicherman die to each other and to the least order of a Sicherman sack
containing the die (Theorems 5–9). For example, there are dice of order pn that lie
in a Sicherman sack of n dice, but in no smaller Sicherman sack.
Later sequels consider variants of these questions. A first variant involves sacks
0
S required to have the same total distribution as that of a sack S of standard dice,
but allowed to consist of dice of orders different from those in S. A second involves
sacks S0 with a different set of values for the totals but with identical probabilities
(when the totals are arranged by size). Fowler and Swift [8] consider these variant
problems when S is a pair of cubical dice. For the first, they show that if S0 is a pair
of dice of orders (k, k0 ), then kk0 = 36, give 2 examples with (k, k0 ) = (2, 18), and
invite the reader to find all others. For the second, they show that if (k, k0 ) = (6, 6),
then all solutions are translates of the standard and Sicherman pairs. The first
two sections of Rosin, Sharobiem, and Swift [16] rehearse the results in [8]. The
third considers sacks S of 2 dice of orders (k, k0 ) with palindromic totals and all
differences of adjacent totals having the same absolute value, and use the structure
of Pythagorean triples to produce an infinite set of examples.
Finally, Brunson and Swift [3] treat the Sicherman variant of the fair totals
problem: which sets of totals can occur with equal probabilities from a relabeled
pair of 6-sided dice. They show that this is always possible if the number of distinct
totals divides 6 and the last section of [16] shows that this number must divide
36. The latter paper then asks when a sack S of n dice of equal order k can be
labeled to give equally likely totals, showing that the answer is positive for all
n when k = n − 1, for infinitely many n when m = 6, and for infinitely many n
when k = 2p = q + 1 with p and q odd primes and both 2 and p generators of
∗
Z/(n − 1)Z , for example, when k equals 14 or 38.
Prior work in problems related to dice
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References
[1] Bancroft H. Brown, Questions and Discussions: Discussion: Probabilities in the Game of “Shooting Craps.”,
Amer. Math. Monthly 26 (1919), no. 8, 351–352, DOI 10.2307/2973388. MR1519357 ←2
[2] Duane M. Broline, Renumbering of the faces of dice, Math. Mag. 52 (1979), no. 5, 312–314, DOI
10.2307/2689786. MR551688 (83b:60013) ←4
[3] Barry W. Brunson and Randall J. Swift, Equally Likely Sums, Math. Spectrum 30 (1998), no. 2, 34–36.
←4
[4] Maurice C. Bryson, Craps with crooked dice, The American Statistician 27 (1973), no. 4, 167–168. ←2
[5] Guantao Chen, M. Bhaskara Rao, and Warren E. Shreve, Can One Load a Set of Dice So That the Sum
Is Uniformly Distributed?, Math. Mag. 70 (1997), no. 3, 204–206. MR1573243 ←1
[6] Ian Morrison and David Swinarski, Computer-assisted calculations and other ancillary material for “Can you play a fair game of craps with a loaded pair of dice?” (2014).
http://faculty.fordham.edu/dswinarski/totaltoparts/. ←2
[7] Edward J. Dudewicz and Robert E. Dann, Equally Likely Dice Sums Do Not Exist, The American
Statistician 26 (1972), no. 4, 41–42. ←1
[8] Brian C. Fowler and Randall J. Swift, Relabeling dice, College Math. J. 30 (1999), no. 3, 204–208, DOI
10.2307/2687599. MR1698572 ←4
[9] Joseph A. Gallian and David J. Rusin, Cyclotomic polynomials and nonstandard dice, Discrete Math. 27
(1979), no. 3, 245–259, DOI 10.1016/0012-365X(79)90161-4. MR541471 (81a:05014) ←4
[10] Martin Gardner, Mathematical games, Sci. Amer. 238 (1978), no. 2, 19–32, DOI
110.1038/scientificamerican0278-19. ←3
[11] William I. Gasarch and Clyde P. Kruskal, When Can One Load a Set of Dice so That the Sum Is Uniformly
Distributed?, Math. Mag. 72 (1999), no. 2, 133–138. MR1573382 ←1
[12] Paul Halmos, Problems for mathematicians, young and old, The Dolciani Mathematical Expositions,
vol. 12, Mathematical Association of America, Washington, DC, 1991. MR1143283 (92j:00009) ←1,
3
[13] J. B. Kelly, Leo Moser, J. H. Wahab, J. V. Finch, and P. R. Halmos, E925, Amer. Math. Monthly 58
(1951), no. 3, 191–192. ←1
[14] Emanuel Parzen, Modern probability theory and its applications, A Wiley Publication in Mathematical
Statistics, John Wiley & Sons Inc., New York, 1960. MR0112166 (22 #3021) ←1
[15] Lewis C. Robertson, Rae Michael Shortt, and Stephen G. Landry, Dice with fair sums, Amer. Math.
Monthly 95 (1988), no. 4, 316–328, DOI 10.2307/2323563. MR935206 (89h:60019) ←1, 3
[16] Amber Rosin, Mary Sharobiem, and Randall Swift, Dice sums, Math. Sci. 33 (2008), no. 2, 99–109.
MR2474736 (2009k:60004) ←4
[17] Armand V. Smith, Some Probability Problems in the Game of “Craps”, The American Statistician 22
(1968), no. 3, 29–30. ←2
[18] Jordan M. Stoyanov, Counterexamples in probability, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons Ltd., Chichester, 1987.
MR930671 (89f:60001) ←3
Online documents referenced
[19] Ian Morrison and David Swinarski, Computer-assisted calculations and other ancillary material for “Can
you play a fair game of craps with a loaded pair of dice?” (2013). In the body of the paper, citations
include the name of a file (where possible as the text of a hyperlink to it). These files are located in
the directory http://faculty.fordham.edu/dswinarski/totaltoparts/ which also contains a
descriptive table of contents toc.htm numbered in parallel with this paper and linking to all these
files. Citations point either to individual files containing the annotated source code and output of a
calculation or other background material, or, for groups of related files, to an anchor in the contents
page at which links to the individual files may be found. ←2