Amusements in Mathematics HTML version

The game has been played with different rules at different periods and places. I give a
copy of the board. Sometimes the diagonal lines are omitted, but this evidently was not
intended to affect the play: it simply meant that the angles alone were thought sufficient
to indicate the points. This is how Strutt, in Sports and Pastimes, describes the game, and
it agrees with the way I played it as a boy:—"Two persons, having each of them nine
pieces, or men, lay them down alternately, one by one, upon the spots; and the business
of either party is to prevent his antagonist from placing three of his pieces so as to form a
row of three, without the intervention of an opponent piece. If a row be formed, he that
made it is at liberty to take up one of his competitor's pieces from any part he thinks most
to his advantage; excepting he has made a row, which must not be touched if he have
another piece upon the board that is not a component part of that row. When all the pieces
are laid down, they are played backwards and forwards, in any direction that the lines
run, but only can move from one spot to another (next to it) at one time. He that takes off
all his antagonist's pieces is the conqueror.
The six educated frogs in the illustration are trained to reverse their order, so that their
numbers shall read 6, 5, 4, 3, 2, 1, with the blank square in its present position. They can
jump to the next square (if vacant) or leap over one frog to the next square beyond (if
vacant), just as we move in the game of draughts, and can go backwards or forwards at
pleasure. Can you show how they perform their feat in the fewest possible moves? It is
quite easy, so when you have done it add a seventh frog to the right and try again. Then
add more frogs until you are able to give the shortest solution for any number. For it can
always be done, with that single vacant square, no matter how many frogs there are.
It has been suggested that this puzzle was a great favourite among the young apprentices
of the City of London in the sixteenth and seventeenth centuries. Readers will have
noticed the curious brass grasshopper on the Royal Exchange. This long-lived creature
escaped the fires of 1666 and 1838. The grasshopper, after his kind, was the crest of Sir
Thomas Gresham, merchant grocer, who died in 1579, and from this cause it has been
used as a sign by grocers in general. Unfortunately for the legend as to its origin, the
puzzle was only produced by myself so late as the year 1900. On twelve of the thirteen
black discs are placed numbered counters or grasshoppers. The puzzle is to reverse their
order, so that they shall read, 1, 2, 3, 4, etc., in the opposite direction, with the vacant disc
left in the same position as at present. Move one at a time in any order, either to the
adjoining vacant disc or by jumping over one grasshopper, like the moves in draughts.
The moves or leaps may be made in either direction that is at any time possible. What are
the fewest possible moves in which it can be done?
Our six educated frogs have learnt a new and pretty feat. When placed on glass tumblers,
as shown in the illustration, they change sides so that the three black ones are to the left
and the white frogs to the right, with the unoccupied tumbler at the opposite end—No. 7.
They can jump to the next tumbler (if unoccupied), or over one, or two, frogs to an