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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.

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170.—­THE CUSHION COVERS.

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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.

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172.—­MRS. SMILEY’S CHRISTMAS PRESENT.

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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.

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There is, I believe, practically only one solution to this puzzle.  The fewest separate squares must be eleven.  The portions must be of the sizes given, the three largest pieces must be arranged as shown, and the remaining group of eight squares may be “reflected,” but cannot be differently arranged.

174.—­THE SQUARES OF BROCADE.

[Illustration:  Diagram 1]

So far as I have been able to discover, there is only one possible solution to fulfil the conditions.  The pieces fit together as in Diagram 1, Diagrams 2 and 3 showing how the two original squares are to be cut.  It will be seen that the pieces A and C have each twenty chequers, and are therefore of equal area.  Diagram 4 (built up with the dissected square No. 5) solves the puzzle, except for the small condition contained in the words, “I cut the