Related papers: A simple solution to Ulam's liar game with one lie
We examine Euclid's lemma that if $p$ is a prime number such that $p | ab$, then $p$ divides at least one of $a$ or $b$. Specifically, we consider the common misapplication of this lemma to numbers that are not prime, as is often made by…
Confidence calibration is essential for making large language models (LLMs) reliable, yet existing training-free methods have been primarily studied under single-answer question answering. In this paper, we show that these methods break…
This short article is aimed at educators and teachers of mathematics.Its goal is simple and direct:to explore some of the basic/elementary properties of proper rational numbers.A proper rational number is a rational which is not an integer.…
User queries are often underspecified and may admit multiple valid interpretations. Rather than silently making assumptions about the user's intent, a helpful assistant should surface such ambiguity by asking a clarifying question. Doing so…
A set $A$ is coarsely computable with density $r \in [0,1]$ if there is an algorithm for deciding membership in $A$ which always gives a (possibly incorrect) answer, and which gives a correct answer with density at least $r$. To any Turing…
Let $(U_n)_{n\in \mathbb{N}}$ be a fixed linear recurrence sequence defined over the integers (with some technical restrictions). We prove that there exist effectively computable constants $B$ and $N_0$ such that for any $b,c\in \mathbb{Z}$…
Alice holds an random variable $X$, and Bob is trying to guess its value by asking questions of the form "is $X=x$?". Alice answers truthfully and the game terminates once Bob guesses correctly. Before the game begins, Bob is allowed to…
A filter oracle for a clutter consists of a finite set $V$ along with an oracle which, given any set $X\subseteq V$, decides in unit time whether or not $X$ contains a member of the clutter. Let $\mathfrak{A}_{2n}$ be an algorithm that,…
Defeasible statements are statements that are likely, or probable, or usually true, but may occasionally be false. Plausible reasoning makes conclusions from statements that are either facts or defeasible statements without using numbers.…
The well-known three distance theorem states that there are at most three distinct gaps between consecutive elements in the set of the first n multiples of any real number. We generalise this theorem to higher dimensions under a suitable…
This paper introduces a strategy in the two envelopes problem that utilizes the prior beliefs of two players about the amount of money that their envelopes can contain. This strategy gives them more information about the decision of…
This paper explores special arrangements of clues in $m \times n$ Numbrix puzzles. The maximum number of clues which fails to define an $m \times n$ puzzle is demonstrated for all $m$ and $n$. In addition, a small upper bound on the minimum…
Users often make ambiguous requests that require clarification. We study the problem of asking clarification questions in an information retrieval setting, where systems often face ambiguous search queries and it is challenging to turn the…
Resolving ambiguities through interaction is a hallmark of natural language, and modeling this behavior is a core challenge in crafting AI assistants. In this work, we study such behavior in LMs by proposing a task-agnostic framework for…
It has been shown that any 9 by 9 Sudoku puzzle must contain at least 17 clues to have a unique solution. This paper investigates the more specific question: given a particular completed Sudoku grid, what is the minimum number of clues in…
Given a finite nonempty sequence of integers S, by grouping adjacent terms it is always possible to write it, possibly in many ways, as S = X Y^k, where X and Y are sequences and Y is nonempty. Choose the version which maximizes the value…
A real number is a rule that, when provided with a rational interval, answers Yes or No depending on if the real number ought to be considered to be in the given interval. Since the goal is to define the real numbers, this can only motivate…
A (continuous) necklace is simply an interval of the real line colored measurably with some number of colors. A well-known application of the Borsuk-Ulam theorem asserts that every $k$-colored necklace can be fairly split by at most $k$…
In this short paper, we give an upper bound for the number of different basic feasible solutions generated by the simplex method for linear programming problems having optimal solutions. The bound is polynomial of the number of constraints,…
We show that there are arbitrarily large sets $S$ of $s$ primes for which the number of solutions to $a+1=c$ where all prime factors of $ac$ lie in $S$ has $\gg \exp( s^{1/4}/\log s)$ solutions.