I wonder how many of my readers, amongst those who have not given any close attention to the elements of geometry, could draw a regular pentagon, or five-sided figure, if they suddenly required to do so.  A regular hexagon, or six-sided figure, is easy enough, for everybody knows that all you have to do is to describe a circle and then, taking the radius as the length of one of the sides, mark off the six points round the circumference.  But a pentagon is quite another matter.  So, as my puzzle has to do with the cutting up of a regular pentagon, it will perhaps be well if I first show my less experienced readers how this figure is to be correctly drawn.  Describe a circle and draw the two lines H B and D G, in the diagram, through the centre at right angles.  Now find the point A, midway between C and B. Next place the point of your compasses at A and with the distance A D describe the arc cutting H B at E. Then place the point of your compasses at D and with the distance D E describe the arc cutting the circumference at F. Now, D F is one of the sides of your pentagon, and you have simply to mark off the other sides round the circle.  Quite simple when you know how, but otherwise somewhat of a poser.

[Illustration]

Having formed your pentagon, the puzzle is to cut it into the fewest possible pieces that will fit together and form a perfect square.

[Illustration]

156.—­THE DISSECTED TRIANGLE.

A good puzzle is that which the gentleman in the illustration is showing to his friends.  He has simply cut out of paper an equilateral triangle—­that is, a triangle with all its three sides of the same length.  He proposes that it shall be cut into five pieces in such a way that they will fit together and form either two or three smaller equilateral triangles, using all the material in each case.  Can you discover how the cuts should be made?

Remember that when you have made your five pieces, you must be able, as desired, to put them together to form either the single original triangle or to form two triangles or to form three triangles—­all equilateral.

157.—­THE TABLE-TOP AND STOOLS.

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.