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The Japanese Ladies An The Carpet

 Answer : If the squares had not to be all the same size, the carpet could be cut in four pieces in any one of the three manners shown.  In each case the two pieces marked A will fit together and form one of the three squares, the other two squares being entire.  But in order to have the squares exactly equal in size, we shall require six pieces, as shown in the larger diagram. No. 1 is a complete square, pieces 4 and 5 will form a second square, and pieces 2, 3, and 6 will form the third—all of exactly the same size. If with the three equal squares we form the rectangle IDBA, then the mean proportional of the two sides of the rectangle will be the side of a square of equal area.  Produce AB to C, making BC equal to BD.  Then place the point of the compasses at E (midway between A and C) and describe the arc AC.  I am showing the quite general method for converting rectangles to squares, but in this particular case we may, of course, at once place our compasses at E, which requires no finding.  Produce the line BD, cutting the arc in F, and BF will be the required side of the square.  Now mark off AG and DH, each equal to BF, and make the cut IG, and also the cut HK from H, perpendicular to ID.  The six pieces produced are numbered as in the diagram on last page. It will be seen that I have here given the reverse method first: to cut the three small squares into six pieces to form a large square.  In the case of our puzzle we can proceed as follows: Make LM equal to half the diagonal ON.  Draw the line NM and drop from L a perpendicular on NM.  Then LP will be the side of all the three squares of combined area equal to the large square QNLO.  The reader can now cut out without difficulty the six pieces, as shown in the numbered square on the last page.

Medieval Brain Teasers