THE TABLE-TOP AND STOOLS.
(
Various Dissection Puzzles)
I have frequently had occasion to show that the published answers to a
great many of the oldest and most widely known puzzles are either quite
incorrect or capable of improvement. I propose to consider the old poser
of the table-top and stools that most of my readers have probably seen
in some form or another in books compiled for the recreation of
childhood.
The story is told that an economical and ingenious schoolmaster once
wished to convert a circular table-top, for which he had no use, into
seats for two oval stools, each with a hand-hole in the centre. He
instructed the carpenter to make the cuts as in the illustration and
then join the eight pieces together in the manner shown. So impressed
was he with the ingenuity of his performance that he set the puzzle to
his geometry class as a little study in dissection. But the remainder of
the story has never been published, because, so it is said, it was a
characteristic of the principals of academies that they would never
admit that they could err. I get my information from a descendant of the
original boy who had most reason to be interested in the matter.
The clever youth suggested modestly to the master that the hand-holes
were too big, and that a small boy might perhaps fall through them. He
therefore proposed another way of making the cuts that would get over
this objection. For his impertinence he received such severe
chastisement that he became convinced that the larger the hand-hole in
the stools the more comfortable might they be.
Now what was the method the boy proposed?
Can you show how the circular table-top may be cut into eight pieces
that will fit together and form two oval seats for stools (each of
exactly the same size and shape) and each having similar hand-holes of
smaller dimensions than in the case shown above? Of course, all the wood
must be used.
Answer:
One object that I had in view when presenting this little puzzle was to
point out the uncertainty of the meaning conveyed by the word "oval."
Though originally derived from the Latin word _ovum_, an egg, yet what
we understand as the egg-shape (with one end smaller than the other) is
only one of many forms of the oval; while some eggs are spherical in
shape, and a sphere or circle is most certainly not an oval. If we speak
of an ellipse--a conical ellipse--we are on safer ground, but here we
must be careful of error. I recollect a Liverpool town councillor, many
years ago, whose ignorance of the poultry-yard led him to substitute the
word "hen" for "fowl," remarking, "We must remember, gentlemen, that
although every cock is a hen, every hen is not a cock!" Similarly, we
must always note that although every ellipse is an oval, every oval is
not an ellipse. It is correct to say that an oval is an oblong
curvilinear figure, having two unequal diameters, and bounded by a curve
line returning into itself; and this includes the ellipse, but all other
figures which in any way approach towards the form of an oval without
necessarily having the properties above described are included in the
term "oval." Thus the following solution that I give to our puzzle
involves the pointed "oval," known among architects as the "vesica
piscis."
[Illustration: THE TWO STOOLS.]
The dotted lines in the table are given for greater clearness, the cuts
being made along the other lines. It will be seen that the eight pieces
form two stools of exactly the same size and shape with similar
hand-holes. These holes are a trifle longer than those in the
schoolmaster's stools, but they are much narrower and of considerably
smaller area. Of course 5 and 6 can be cut out in one piece--also 7 and
8--making only _six pieces_ in all. But I wished to keep the same number
as in the original story.
When I first gave the above puzzle in a London newspaper, in
competition, no correct solution was received, but an ingenious and
neatly executed attempt by a man lying in a London infirmary was
accompanied by the following note: "Having no compasses here, I was
compelled to improvise a pair with the aid of a small penknife, a bit of
firewood from a bundle, a piece of tin from a toy engine, a tin tack,
and two portions of a hairpin, for points. They are a fairly serviceable
pair of compasses, and I shall keep them as a memento of your puzzle."
Informational
