# Busy Beaver Scores and Alphabet Size

## TL;DR

This paper proves that increasing alphabet size in the Busy Beaver game can lead to higher activity and productivity, especially with larger state counts, extending Harland's conjecture to non-binary alphabets.

## Contribution

It confirms Harland's conjecture for large state counts and demonstrates how increasing alphabet size enhances activity and productivity in Busy Beaver machines.

## Key findings

- Increasing alphabet size from two to three increases activity.
- Larger alphabet sizes can lead to higher productivity.
- A construction allows increasing activity with a larger alphabet depending on states.

## Abstract

We investigate the Busy Beaver Game introduced by Rado (1962) generalized to non-binary alphabets. Harland (2016) conjectured that activity (number of steps) and productivity (number of non-blank symbols) of candidate machines grow as the alphabet size increases. We prove this conjecture for any alphabet size under the condition that the number of states is sufficiently large. For the measure activity we show that increasing the alphabet size from two to three allows an increase. By a classical construction it is even possible to obtain a two-state machine increasing activity and productivity of any machine if we allow an alphabet size depending on the number of states of the original machine. We also show that an increase of the alphabet by a factor of three admits an increase of activity.