Henry Ernest Dudeney/Puzzles and Curious Problems/304 - Grasshoppers' Quadrille/Solution

by : $304$

 * Grasshoppers' Quadrille

Solution
The counters can be exchanged in $120$ moves.

Proof
Consider the central column containing $3$ white and $3$ black counters.

These can be made to change places in $15$ moves.

Number the $7$ squares downwards from $1$ to $7$.

Now play:
 * $3 - 4$, $5 - 3$, $6 - 5$, $4 - 6$, $2 - 4$, $1 - 2$, $3 - 1$, $5 - 3$, $7 - 5$, $6 - 7$, $4 - 6$, $2 - 4$, $3 - 2$, $5 - 3$, $4 - 5$

of which $6$ are simple moves, and $9$ are jumps.

Now there are $7$ horizontal rows of $3$ white and $3$ black counters, if we exclude that central column.

Each of these can be similarly interchanged in $15$ moves.

For each row, this can be done at any stage where there is a vacant space in the central column.

For example, the central row can be done straight away, as the central space starts empty.

So, as there are $7$ rows and $1$ column, the counters can be exchanged in $8 \times 15 = 120$ moves.