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Amusements in Mathematics

Points And Lines Problems
"Line upon line, line upon line; here a little and there a little."—Isa. xxviii. 10.
What are known as "Points and Lines" puzzles are found very interesting by many
people. The most familiar example, here given, to plant nine trees so that they shall form
ten straight rows with three trees in every row, is attributed to Sir Isaac Newton, but the
earliest collection of such puzzles is, I believe, in a rare little book that I possess—
published in 1821—Rational Amusement for Winter Evenings, by John Jackson. The
author gives ten examples of "Trees planted in Rows."
These tree-planting puzzles have always been a matter of great perplexity. They are real
"puzzles," in the truest sense of the word, because nobody has yet succeeded in finding a
direct and certain way of solving them. They demand the exercise of sagacity, ingenuity,
and patience, and what we call "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.
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.
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