Credit for this problem goes to the 51st IMO, problem 5. Thanks!

Coin Towers

In each of six boxes $B_1$, $B_2$, $B_3$, $B_4$, $B_5$, $B_6$ there is initially one coin. There are two types of operation allowed:

Type 1: Choose a nonempty box $B_j$ with $1 \leq j \leq 5$. Remove one coin from $B_j$ and add two coins to $B_{j+1}$.

Type 2: Choose a nonempty box $B_k$ with $1 \leq k \leq 4$. Remove one coin from $B_k$ and exchange the contents of (possibly empty) boxes $B_{k+1}$ and $B_{k+2}$.

Warm up problem: Using only operations of Type 1, what’s the most coins you can have in total across all 6 boxes?

Main problem: Using operations of either type, what’s the most coins you can have in total across all 6 boxes?

Solution