Fano Geometry and Slow Coupon Collecting

We study the coupon collector's problem in a generalized setting where each draw reveals a fixed number of coupons and the sampling mechanism is required to be \emph{fair}, meaning that every coupon appears with the same frequency among the admissible draws.

Grunbaum and Yaakobi conjectured that, among all fair mechanisms with fixed parameters, the fully random model maximizes the expected time to complete coverage.

We disprove this conjecture by exhibiting explicit counterexamples arising from finite geometry.

In particular, we show that the line set of the Fano plane yields a fair mechanism whose expected coverage time exceeds that of the full model.

Further exact and computational results are obtained for projective planes of higher order.

In addition, we analyze a simple infinite family of fair mechanisms, the star mechanism, for which the expected coverage time admits a closed form.

Depending on the scaling regime, this mechanism can be asymptotically slower or faster than the full model, showing that no universal extremality principle holds for fair mechanisms without additional structural assumptions.