In the Amemiya-Ando problem 3 is enough

In any infinite dimensional Hilbert space $\mathcal{H}$ there exist orthogonal projections $Q_1$, $Q_2$ and $Q_3$, such that a sequence $(P_n\mathrm{.}\mathrm{.}\mathrm{.}{P}_1(x))$ diverges in norm for some $P_1,P_2,\mathrm{.}\mathrm{.}\mathrm{.}\in \{Q_1,Q_2,Q_3\}$ and $x\in \mathcal{H}$.