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Mastermind with a linear number of queries
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- Journal:
- Combinatorics, Probability and Computing / Volume 33 / Issue 2 / March 2024
- Published online by Cambridge University Press:
- 08 November 2023, pp. 143-156
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- Article
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Since the 1960s Mastermind has been studied for the combinatorial and information-theoretical interest the game has to offer. Many results have been discovered starting with Erdős and Rényi determining the optimal number of queries needed for two colours. For
$k$ colours and 
$n$ positions, Chvátal found asymptotically optimal bounds when 
$k \le n^{1-\varepsilon }$. Following a sequence of gradual improvements for 
$k\geq n$ colours, the central open question is to resolve the gap between 
$\Omega (n)$ and 
$\mathcal{O}(n\log \log n)$ for 
$k=n$. In this paper, we resolve this gap by presenting the first algorithm for solving 
$k=n$ Mastermind with a linear number of queries. As a consequence, we are able to determine the query complexity of Mastermind for any parameters 
$k$ and 
$n$.
