Henry Ernest Dudeney/Modern Puzzles/153 - Transferring the Counters/Solution

by : $153$

 * Transferring the Counters

Solution
First we establish that if you have $n$ empty squares, you can place a pile of $n$ counters on one of those squares from another square in $2 n - 1$ moves.

This is trivially done by:
 * placing each of the $n$ counters on an arbitrary empty square, taking $n$ moves
 * placing $n - 1$ of these counters on the $n$th counter in numerical order.

So:

Make a pile of $5$ counters, numbers $1$ to $5$, on $B$ in $9$ moves.

Make a pile of $4$ counters, numbers $6$ to $9$, on $C$ in $7$ moves.

Make a pile of $3$ counters, numbers $10$ to $12$, on $D$ in $5$ moves.

Make a pile of $2$ counters, numbers $13$ and $14$, on $E$ in $3$ moves.

Place number $15$ on $F$ in one move.

Place $13$ and $14$ on $F$ in $3$ moves.

Place $10$ to $12$ on $F$ in $5$ moves.

Place $6$ to $9$ on $F$ in $7$ moves.

Place $1$ to $5$ on $F$ in $9$ moves.

Total: $49$ moves.