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

Modern Puzzles by Henry Ernest Dudeney: $153$

Transferring the Counters

Divide a sheet of paper into six compartments, as shown in the diagram,

and place a pile of $15$ counters, numbered consecutively $1$, $2$, $3$, $\ldots$, $15$ downwards, in compartment $A$.

The puzzle is to transfer the complete pile, in the fewest possible moves, to compartment $F$.

You can move the counters one at a time to any compartment,

but may never place a counter on one that bears a smaller number than itself.

Thus, if you place $1$ on $B$ and $2$ on $C$, you can then place $1$ on $2$, but not $2$ on $1$.

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.

Sources

• 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $153$. -- Transferring the Counters
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $370$. Transferring the Counters