Choosing 1 of N with and without lucky numbers

TL;DR

This paper explores the number of fair coin tosses needed to uniformly select one of n options, revealing deep number-theoretic connections, optimal schemes, and fractal structures, with implications for lotteries and simulations.

Contribution

It introduces a new bit-efficient scheme for N-way selection, proves its optimality, and characterizes the expected number of tosses with fractal and number-theoretic insights.

Findings

Derived the expected number of coin tosses e[n]

Proved the optimality of the proposed scheme

Identified fractal structure in the expected value function

Abstract

How many fair coin tosses to choose 1 of $n$ options with uniform probability? Although a probability problem, the solution is essentially number-theoretic, with special roles for Mersenne numbers, Fermat numbers, and the haupt exponent. We propose a bit-efficient scheme, prove optimality, derive the expected number of coin tosses $e[n]$, characterize its fractal structure, and develop sharp upper and lower bounds, both discrete and continuous. A minor but noteworthy corollary, with real-world examples, is that any lottery or simulation with finite budget of random bits will have a predictable pattern of lucky and unlucky numbers.