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Joe Buhler

Publications and source records attributed to Joe Buhler.

6 recordsLinked to original sources
arXiv2021

On Levine's notorious hat puzzle

The Levine hat game requires $n$ players, each wearing an infinite random stack of black and white hats, to guess the location of a black hat on their own head seeing only the hats worn by all the other players. They are allowed a strategy session before the game, but no further communication. The players collectively win if and only if all their guesses are correct. In this paper we give an overview of what is known about strategies for this game, including an extended discussion of the case with $n = 2$ players (and a conjecture for an optimal strategy in this case). We also prove that $V_n$, the optimal value of the joint success probability in the $n$-player game, is a strictly decreasing function of $n$.

math.CO↗
arXiv2019

Littlewood Polynomials, Spectral-Null Codes, and Equipowerful Partitions

Let $[n]$ denote $\{0,1, ... , n-1\}$. A polynomial $f(x) = \sum a_i x^i$ is a Littlewood polynomial (LP) of length $n$ if the $a_i$ are $\pm 1$ for $i \in [n]$, and $a_i = 0$ for $i \ge n$. Such an LP is said to have order $m$ if it is divisible by $(x-1)^m$. The problem of finding the set $L_m$ of lengths of LPs of order $m$ is equivalent to finding the lengths of spectral-null codes of order $m$, and to finding $n$ such that $[n]$ admits a partition into two subsets whose first $m$ moments are equal. Extending the techniques and results of Boyd and others, we completely determine $L_7$ and $L_8$ and prove that 192 is the smallest element of $L_9$. Our primary tools are the use of carefully targeted searches using integer linear programming (both to find LPs and to disprove their existence for specific $n$ and $m$), and an unexpected new concept (that arose out of observed symmetry properties of LPs) that we call "regenerative pairs," which produce infinite arithmetic progressions in $L_m$. We prove that for $m \le$ 8, whenever there is an LP of length $n$ and order $m$, there is one of length $n$ and order $m$ that is symmetric (resp.~antisymmetric) if m is even (resp.~odd).

math.NT↗
arXiv2010

Origami rings

Motivated by a question in origami, we consider sets of points in the complex plane constructed in the following way. Let $L_α(p)$ be the line in the complex plane through $p$ with angle $α$ (with respect to the real axis). Given a fixed collection $U$ of angles, let $\RU$ be the points that can be obtained by starting with $0$ and $1$, and then recursively adding intersection points of the form $L_α(p) \cap L_β(q)$, where $p, q$ have been constructed already, and $α, β$ are distinct angles in $U$. Our main result is that if $U$ is a group with at least three elements, then $\RU$ is a subring of the complex plane, i.e., it is closed under complex addition and multiplication. This enables us to answer a specific question about origami folds: if $n \ge 3$ and the allowable angles are the $n$ equally spaced angles $kπ/n$, $0 \le k < n$, then $\RU$ is the ring $\Z[ζ_n]$ if $n$ is prime, and the ring $\Z[1/n,ζ_{n}]$ if $n$ is not prime, where $ζ_n := \exp(2πi/n)$ is a primitive $n$-th root of unity.

math.CO↗
arXiv2010

Hypercube orientations with only two in-degrees

We consider the problem of orienting the edges of the $n$-dimensional hypercube so only two different in-degrees $a$ and $b$ occur. We show that this can be done, for two specified in-degrees, if and only if an obvious necessary condition holds. Namely, there exist non-negative integers $s$ and $t$ so that $s+t=2^n$ and $as+bt=n2^{n-1}$. This is connected to a question arising from constructing a strategy for a "hat puzzle."

math.CO↗
arXiv2003

Symmetric functions and the phase problem in crystallography

The calculation of crystal structure from X-ray diffraction data requires that the phases of the ``structure factors'' (Fourier coefficients) determined by scattering be deduced from the absolute values of those structure factors. Motivated by a question of Herbert Hauptman, we consider the problem of determining phases by direct algebraic means in the case of crystal structures with $n$ equal atoms in the unit cell, with $n$ small. We rephrase the problem as a question about multiplicative invariants for a particular finite group action. We show that the absolute values form a generating set for the field of invariants of this action, and consider the problem of making this theorem constructive and practical; the most promising approach for deriving explicit formulas uses SAGBI bases.

math.AC↗