Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 224 / 3 / b / Solution

Regard the alphabet as the cyclic group $\mathbb{Z}/m\mathbb{Z}$ and put $E=X-Y$. For each $y$, subtraction by $y$ is a bijection, soBy concavity of $\varphi$, its monotonicity in the distortion allowance, and $\mathbb{E}{d}_Y=\mathbb{P}(X\ne Y)\le d$,Therefore