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. [Illustration] 169.--A TANGRAM PARADOX. The diagrams will show how the figures are constructed--each with the seven Tangrams. It will be noticed that in both cases the head, hat, and arm are precisely alike, and the width at the base of the body the same. But this body contains four pieces in the first case, and in the second design only three. The first is larger than the second by exactly that narrow strip indicated by the dotted line between A and B. This strip is therefore exactly equal in area to the piece forming the foot in the other design, though when thus distributed along the side of the body the increased dimension is not easily apparent to the eye. [Illustration] 170.--THE CUSHION COVERS. [Illustration] The two pieces of brocade marked A will fit together and form one perfect square cushion top, and the two pieces marked B will form the other. 171.--THE BANNER PUZZLE. The illustration explains itself. Divide the bunting into 25 squares (because this number is the sum of two other squares--16 and 9), and then cut along the thick lines. The two pieces marked A form one square, and the two pieces marked B form the other. [Illustration] 172.--MRS. SMILEY'S CHRISTMAS PRESENT. [Illustration] [Illustration] The first step is to find six different square numbers that sum to 196. For example, 1 + 4 + 25 + 36 + 49 + 81 = 196; 1 + 4 + 9 + 25 + 36 + 121 = 196; 1 + 9 + 16 + 25 + 64 + 81 = 196. The rest calls for individual judgment and ingenuity, and no definite rules can be given for procedure. The annexed diagrams will show solutions for the first two cases stated. Of course the three pieces marked A and those marked B will fit together and form a square in each case. The assembling of the parts may be slightly varied, and the reader may be interested in finding a solution for the third set of squares I have given. 173.--MRS. PERKINS'S QUILT. The following diagram shows how the quilt should be constructed. [Illustration]