Lower bounds on the M\"{u}nchhausen problem

TL;DR

This paper establishes new lower bounds for the M"{u}nchhausen problem, demonstrating that for infinitely many n, the minimal number of weighings exceeds the trivial logarithmic bound, highlighting the problem's complexity.

Contribution

It provides the first nontrivial lower bounds for the M"{u}nchhausen problem, showing that B(n) often surpasses the simple log_3(n) estimate.

Findings

There are infinitely many n where B(n) > ceil(log_3 n).

The number of n values with B(n) ≠ ceil(log_3 n) is unbounded as k increases.

The results indicate B(n) can significantly exceed the trivial lower bound.

Abstract

"The Baron's omni-sequence", B(n), first defined by Khovanova and Lewis (2011), is a sequence that gives for each n the minimum number of weighings on balance scales that can verify the correct labeling of n identically-looking coins with distinct integer weights between 1 gram and n grams. A trivial lower bound on B(n) is log_3(n), and it has been shown that B(n) is log_3(n) + O(log log n). In this paper we give a first nontrivial lower bound to the M\"{u}nchhausen problem, showing that there is an infinite number of n values for which B(n) does not equal ceil(log_3 n). Furthermore, we show that if N(k) is the number of n values for which k = ceil(log_3 n) and B(n) does not equal k, then N(k) is an unbounded function of k.