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Amusements in Mathematics

Of course a half-square rectangle is the same as a double square, or two equal squares
joined together. Therefore, if you want to solve the puzzle of cutting a Greek cross into
four pieces to form two separate squares of the same size, all you have to do is to
continue the short cut in Fig. 38 right across the cross, and you will have four pieces of
the same size and shape. Now divide Fig. 37 into two equal squares by a horizontal cut
midway and you will see the four pieces forming the two squares.
Cut a Greek cross into five pieces that will form two separate squares, one of which shall
contain half the area of one of the arms of the cross. In further illustration of what I have
already written, if the two squares of the same size A B C D and B C F E, in Fig. 41, are
cut in the manner indicated by the dotted lines, the four pieces will form the large square
A G E C. We thus see that the diagonal A C is the side of a square twice the size of A B
C D. It is also clear that half the diagonal of any square is equal to the side of a square of
half the area. Therefore, if the large square in the diagram is one of the arms of your
cross, the small square is the size of one of the squares required in the puzzle.
The solution is shown in Figs. 42 and 43. It will be seen that the small square is cut out
whole and the large square composed of the four pieces B, C, D, and E. After what I have
written, the reader will have no difficulty in seeing that the square A is half the size of
one of the arms of the cross, because the length of the diagonal of the former is clearly
the same as the side of the latter. The thing is now self-evident. I have thus tried to show
that some of these puzzles that many people are apt to regard as quite wonderful and
bewildering, are really not difficult if only we use a little thought and judgment. In
conclusion of this particular subject I will give four Greek cross puzzles, with detached
solutions.
142.—THE SILK PATCHWORK.
The lady members of the Wilkinson family had made a simple patchwork quilt, as a small
Christmas present, all composed of square pieces of the same size, as shown in the
illustration. It only lacked the four corner pieces to make it complete. Somebody pointed
out to them that if you unpicked the Greek cross in the middle and then cut the stitches
along the dark joins, the four pieces all of the same size and shape would fit together and
form a square. This the reader knows, from the solution in Fig. 39, is quite easily done.
But George Wilkinson suddenly suggested to them this poser. He said, "Instead of
picking out the cross entire, and forming the square from four equal pieces, can you cut
out a square entire and four equal pieces that will form a perfect Greek cross?" The
puzzle is, of course, now quite easy.
143.—TWO CROSSES FROM ONE.
Cut a Greek cross into five pieces that will form two such crosses, both of the same size.
The solution of this puzzle is very beautiful.
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