# Why Byzantine Consensus Requires 3f + 1 Processes ## Metadata **Status**:: #x **Zettel**:: #zettel/fleeting **Created**:: [[2026-01-19]] ## Synopsis Use - $n$: number total processes - $f$: number of Byzantine faulty processes - $q$: quorum size Any two quorums must contain at least a common correct process: $ 2q - n > f\tag{1} \implies f + n < 2q $ > $2q - n$ is the number of common processes between any two quorums. All correct processes must be able to form a quorum to ensure liveness: $ n - f \ge q \implies 2q \le 2(n-f) \tag{2} $ From (1) and (2): $ \begin{array}{ll} & f + n < 2q <= 2(n - f) \\ \implies & f + n < 2(n - f) \\ \implies & n > 3f \\ \end{array} $