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We determine the maximum number of induced cycles that can be contained in agraph on $n\ge n_0$ vertices, and show that there is a unique graph thatachieves this maximum. This answers a question of Tuza. We also determine themaximum number of odd or even cycles that can be contained in a graph on $n\gen_0$ vertices and characterise the extremal graphs. This resolves a conjectureof Chv\'atal and Tuza from 1988.

Maximising the number of induced cycles in a graph