arXiv subjects

Soham Samanta

Publications and source records attributed to Soham Samanta.

6 recordsLinked to original sources

arXiv2026

Self-Referential Tests

We study self-referential multiple-choice tests with the question: \emph{How many correct answer choices are there?} The answer choices are positive integers. A value $a$ is called \emph{valid} if it occurs exactly $a$ times among answer choices. The \emph{cost} of a test is the sum of all answer choices, linking the problem to integer partitions. Using this framework, we define solvable and $k$-solvable tests and derive generating functions that enumerate them by cost, number of distinct valid values, and number of options. We also investigate extremal questions, including minimum costs and the maximum possible number of valid values. The paper was inspired by a puzzle from \emph{Mathematical Puzzles and Curiosities}.

math.GM↗

arXiv2025

On graphs with equal and different Kromatic symmetric functions

The Kromatic symmetric function (KSF) $\overline{X}_G$ of a graph $G$ is a $K$-analogue introduced by Crew, Pechenik, and Spirkl in arXiv:2301.02177 of Stanley's chromatic symmetric function (CSF) $X_G$. The KSF is known to distinguish some pairs of graphs with the same CSF. The first author showed in arXiv:2403.15929 and arXiv:2502.21285 that the number of copies in $G$ of certain induced subgraphs can be determined given $\overline{X}_G$, and conjectured that $\overline{X}_G$ distinguishes all graphs. We disprove that conjecture by finding four pairs of 8-vertex graphs with equal KSF, as well as giving several ways to use existing graph pairs with equal KSF to construct larger graph pairs that also have equal KSF. On the other hand, we show that many of the graph pairs from the constructions of Orellana and Scott in arXiv:1308.6005 and of Aliste-Prieto, Crew, Spirkl, and Zamora in arXiv:2007.11042 of graphs with the same CSF are distinguished by the KSF, thus also giving some new examples of cases where the KSF is a stronger invariant than the CSF.

math.CO↗

arXiv2025

Mathematics of Gozinta Boxes

We study the geometric aspects of the magic trick called Gozinta Boxes. We generalize Gozinta Boxes to other dimensions, and we show that in three and higher dimensions, the maximum number of boxes is 3, and in two dimensions, the maximum is 4. We discuss other properties of Gozinta Boxes and provide a plethora of examples.

math.GM↗

arXiv2025

Chip-Firing on Infinite $k$-ary Trees

We use an infinite $k$-ary tree with a self-loop at the root as our underlying graph. We consider a chip-firing process starting with $N$ chips at the root. We describe the stable configurations. We calculate the number of fires for each vertex and the total number of fires. We study a sequence of the number of root fires for a given $k$ as a function of $N$ and study its properties. We do the same for the total number of fires.

math.CO↗

arXiv2024

Fibonacci Partial Sums Tricks

The following magic trick is at the center of this paper. While the audience writes the first ten terms of a Fibonacci-like sequence (the sequence following the same recursion as the Fibonacci sequence), the magician calculates the sum of these ten terms very fast by multiplying the 7th term by 11. This trick is based on the divisibility properties of partial sums of Fibonacci-like sequences. We find the maximum Fibonacci number that divides the sum of the Fibonacci numbers 1 through $n$. We discuss the generalization of the trick for other second-order recurrences. We show that a similar trick exists for Pell-like sequences and does not exist for Jacobhstal-like sequences.

math.HO↗

arXiv2024

Fibonometry and Beyond

In 2013, Conway and Ryba wrote a fascinating paper called Fibonometry. The paper, as one might guess, is about the connection between Fibonacci numbers and trigonometry. We were fascinated by this paper and looked at how we could generalize it. We discovered that we weren't the first. In this paper, we describe our journey and summarize the results.

math.HO↗