"luck" is also sometimes of service. Perhaps
some day a genius will discover the key to the whole mystery. Remember
that the trees must be regarded as mere points, for if we were allowed
to make our trees big enough we might easily "fudge" our diagrams and
get in a few extra straight rows that were more apparent than real.

[Illustration]

206.--THE KING AND THE CASTLES.

There was once, in ancient times, a powerful king, who had eccentric
ideas on the subject of military architecture. He held that there was
great strength and economy in symmetrical forms, and always cited the
example of the bees, who construct their combs in perfect hexagonal
cells, to prove that he had nature to support him. He resolved to build
ten new castles in his country all to be connected by fortified walls,
which should form five lines with four castles in every line. The royal
architect presented his preliminary plan in the form I have shown. But
the monarch pointed out that every castle could be approached from the
outside, and commanded that the plan should be so modified that as many
castles as possible should be free from attack from the outside, and
could only be reached by crossing the fortified walls. The architect
replied that he thought it impossible so to arrange them that even one
castle, which the king proposed to use as a royal residence, could be so
protected, but his majesty soon enlightened him by pointing out how it
might be done. How would you have built the ten castles and
fortifications so as best to fulfil the king's requirements? Remember
that they must form five straight lines with four castles in every line.

[Illustration]

207.--CHERRIES AND PLUMS.

[Illustration]

The illustration is a plan of a cottage as it stands surrounded by an
orchard of fifty-five trees. Ten of these trees are cherries, ten are
plums, and the remainder apples. The cherries are so planted as to form
five straight lines, with four cherry trees in every line. The plum
trees are also planted so as to form five straight lines with four plum
trees in every line. The puzzle is to show which are the ten cherry
trees and which are the ten plums. In order that the cherries and plums
should have the most favourable aspect, as few as possible (under the
conditions) are planted on the north and east sides of the orchard. Of
course in picking out a group of ten trees (cherry or plum, as the case
may be) you ignore all intervening trees. That is