Related papers: Odd One Out
Two Turing Machines may be able to answer questions about each other that they cannot answer about themselves.
In the 80th anniversary book for Alex M\"uller I wrote a story of our scientific collaboration, Shared Fascinations. This time I will be more personal, about the human side of our collaboration and encounters, while also referring to…
Here, we present a variant of the sliding coins game. Two coins are placed on distinct squares of a semi-infinite linear board with squares numbered $0, 1, 2, dots, $. Two players take turns and move a coin to a lower unoccupied square.…
The generation of humor is an under-explored and challenging problem. Previous works mainly utilize templates or replace phrases to generate humor. However, few works focus on freer forms and the background knowledge of humor. The…
In the 2013 Advent calender of the Berlin Mathematics Research Center MATH+, Gerhard Woeginger presents a novel hat problem with an uncommon initial announcement. Although the information given is insufficient for the hat bearers to learn…
This is a collection of open problems in geometry that I think of as puzzles: they stick to my brain -- I see many grips, but no spare hands. Puzzle-charm is the only criterion for including a problem here; importance is ignored.
Mathematicians had little idea whether the easy-to-state union-closed conjecture was true or false even after $40$ years. However, last winter saw a surge of interest in the conjecture and its variants, initiated by the contribution of a…
The goal of this note is to construct, on many manifolds, non-trivial concordances from the identity to itself. This produces counterexamples to a recent conjecture by Botvinnik.
Here I share a few notes I used in various course lectures, talks, etc. Some may be just calculations that in the textbooks are more complicated, scattered, or less specific; others may be simple observations I found useful or curious.
The answers people give when asked to 'think of the unexpected' for everyday event scenarios appear to be more expected than unexpected. There are expected unexpected outcomes that closely adhere to the given information in a scenario,…
Inductive inference is a recursion-theoretic theory of learning, first developed by E. M. Gold (1967). This paper surveys developments in probabilistic inductive inference. We mainly focus on finite inference of recursive functions, since…
We consider a class of self-interacting random walks in deterministic or random environments, known as excited random walks or cookie walks, on the d-dimensional integer lattice. The main purpose of this paper is two-fold: to give a survey…
A cohomology theory for "odd polygon" relations -- algebraic imitations of Pachner moves in dimensions 3, 5, ... -- is constructed. Manifold invariants based on polygon relations and nontrivial polygon cocycles are proposed. Example…
The Collatz conjecture is a famous math problem that was introduced by Lothar Collatz in 1937, and nobody has yet succeeded in proving or disproving it. In this article, I will analyze this problem with a new approach and I will discuss my…
This article introduces a pedagogical method for {\it solving combinatorial problems} that frequently involve structures that are unfamiliar or less familiar. Indeed, an indirect method has been proposed in order to evade any possible…
Ron Graham's Sequence is a surprising bijection from non-negative integers to non-negative, non-prime integers that was introduced by Ron Graham in the June 1986 "Problems" column of $\textit{Mathematics Magazine}$, and which later appeared…
Scattered factor (circular) universality was firstly introduced by Barker et al. in 2020. A word $w$ is called $k$-universal for some natural number $k$, if every word of length $k$ of $w$'s alphabet occurs as a scattered factor in $w$; it…
Incomputability as a mathematical notion arose from work of Alan Turing and Alonzo Church in the 1930s. Like Turing himself, it attracted less attention than it deserved beyond the confines of mathematics. Today our experiences in computer…
An origin is often an intriguing issue. It becomes doubly intriguing when the logical form of thinking is considered. In this paper we will investigate exactly that: we will conjecture on the origin of basic instruments of logical thinking.…
We help Alice play a certain "convergence game" against Bob and win the prize, which is a constructive solution to a problem by Erd\H{o}s and Graham, posed in their 1980 book on open questions in combinatorial number theory. Namely, after…