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

Puzzles and Curious Problems by Henry Ernest Dudeney: $304$

It is required to make the white men change places with the black men in the fewest possible moves.

There is no diagonal play, nor are there captures.

The white men can only move to the right or downwards, and the black men to the left or upwards,

but they may leap over one of the opposite colour, as in draughts.

The counters can be exchanged in $120$ moves.

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.

$\blacksquare$