through the card along the short portions of B B and C C that are not dotted.  Next turn the card over and cut half through along the short lines on B B and C C at the places that are immediately beneath the dotted lines on the upper side.  With a little careful separation of the parts with the penknife, the cardboard may now be divided into two interlacing ladder-like portions, as shown in Fig. 2; and if you cut away all the shaded parts you will get the chain, cut solidly out of the cardboard, without any join, as shown in the illustrations on page 40.

It is an interesting variant of the puzzle to cut out two keys on a ring—­in the same manner without join.

[Illustration]

164.—­THE POTATO PUZZLE.

As many as twenty-two pieces may be obtained by the six cuts.  The illustration shows a pretty symmetrical solution.  The rule in such cases is that every cut shall intersect every other cut and no two intersections coincide; that is to say, every line passes through every other line, but more than two lines do not cross at the same point anywhere.  There are other ways of making the cuts, but this rule must always be observed if we are to get the full number of pieces.

The general formula is that with n cuts we can always produce (n(n + 1) + 1)/2 .  One of the problems proposed by the late Sam Loyd was to produce the maximum number of pieces by n straight cuts through a solid cheese.  Of course, again, the pieces cut off may not be moved or piled.  Here we have to deal with the intersection of planes (instead of lines), and the general formula is that with n cuts we may produce ((n — 1)n(n + 1))/6 + n + 1 pieces.  It is extremely difficult to “see” the direction and effects of the successive cuts for more than a few of the lowest values of n.

165.—­THE SEVEN PIGS.

The illustration shows the direction for placing the three fences so as to enclose every pig in a separate sty.  The greatest number of spaces that can be enclosed with three straight lines in a square is seven, as shown in the last puzzle.  Bearing this fact in mind, the puzzle must be solved by trial.

[Illustration:  THE SEVEN PIGS.]

166.—­THE LANDOWNER’S FENCES.

Four fences only are necessary, as follows:—­

[Illustration]

167.—­THE WIZARD’S CATS.

The illustration requires no explanation.  It shows clearly how the three circles may be drawn so that every cat has a separate enclosure, and cannot approach another cat without crossing a line.

[Illustration:  THE WIZARDS’ CATS.]

168.—­THE CHRISTMAS PUDDING.

The illustration shows how the pudding may be cut into two parts of exactly the same size and shape.  The lines must necessarily pass through the points A, B, C, D, and E. But, subject to this condition, they may be varied in an infinite number of ways.  For example, at a point midway between A and the edge, the line may be completed in an unlimited number of ways (straight or crooked), provided it be exactly reflected from E to the opposite edge.  And similar variations may be introduced at other places.