Solution of an odds inversion problem

Robert K. Moniot
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 3, Pages 460–470
DOI: 10.7546/nntdm.2025.31.3.460-470
Full paper (PDF, 228 Kb)

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Authors and affiliations

Robert K. Moniot
Department of Computer and Information Science, Fordham University
113 W. 60th St, New York, NY 10023, USA

Abstract

Consider the problem of determining the possible numbers of balls of two different colors in an urn such that if two are drawn out at random, the odds that they are different colors are a given value. We present a general solution of this problem for all odds from nil to certainty. The solution methods use relatively simple concepts from number theory such as modular inverses and the Pell equation. We find upper bounds on the number of solutions and the magnitude of solutions for those cases that have at most a finite number of solutions. We also define solution classes for cases that have an infinite number of solutions, and identify cases having a determinate number of solution classes.

Keywords

  • Pell equation
  • Modular inverses
  • Combinatorial probability
  • Quadratic Diophantine equations
  • Linear congruences
  • Odds inversion

2020 Mathematics Subject Classification

  • 11A07
  • 11D09
  • 11D45
  • 60C05

References

  1. Hilmer, K., Jin, A., Lycan, R., & Ponomarenko, V. (2023). The elliptical case of an odds inversion problem. Involve, 16(3), 431–452.
  2. Hilmer, K., Lycan, R., & Ponomarenko, V. (2022). Odds inversion problem with
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  3. Hua, L. (1982). Introduction to Number Theory (translated from the Chinese by Peter Shiu). Springer-Verlag Berlin Heidelberg.
  4. Moniot, R. K. (2021). Solution of an odds inversion problem. The American Mathematical Monthly, 128(2), 140–149.
  5. Nagell, T. (1964). Introduction to Number Theory (2nd ed.). Chelsea Publishing Co., New York.
  6. National Museum of Mathematics, New York City (2017). Fifty-Fifty. Varsity
    Math, Week 117. Available online at: https://momath.org/all-events/ongoing-programs/varsity-math/varsity-math-week-117/.

Manuscript history

  • Received: 26 November 2024
  • Revised: 1 August 2025
  • Accepted: 2 August 2025
  • Online First: 4 August 2025

Copyright information

Ⓒ 2025 by the Author.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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Cite this paper

Moniot, R. K. (2025). Solution of an odds inversion problem. Notes on Number Theory and Discrete Mathematics, 31(3), 460-470, DOI: 10.7546/nntdm.2025.31.3.460-470.

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