Amusements in Mathematics HTML version

"Are they not exquisite?" said my friend. "They were brought to me by a cousin who has
just returned from India. Now, I want you to give me a little assistance. You see, I have
decided to join them together so as to make one large square cushion-cover. How should
I do this so as to mutilate the material as little as possible? Of course I propose to make
my cuts only along the lines that divide the little chequers."
I cut the two squares in the manner desired into four pieces that would fit together and
form another larger square, taking care that the pattern should match properly, and when I
had finished I noticed that two of the pieces were of exactly the same area; that is, each of
the two contained the same number of chequers. Can you show how the cuts were made
in accordance with these conditions?
A lady was presented, by two of her girl friends, with the pretty pieces of silk patchwork
shown in our illustration. It will be seen that both pieces are made up of squares all of the
same size—one 12x12 and the other 5x5. She proposes to join them together and make
one square patchwork quilt, 13x13, but, of course, she will not cut any of the material—
merely cut the stitches where necessary and join together again. What perplexes her is
this. A friend assures her that there need be no more than four pieces in all to join up for
the new quilt. Could you show her how this little needlework puzzle is to be solved in so
few pieces?
The diagram herewith represents two separate pieces of linoleum. The chequered pattern
is not repeated at the back, so that the pieces cannot be turned over. The puzzle is to cut
the two squares into four pieces so that they shall fit together and form one perfect square
10×10, so that the pattern shall properly match, and so that the larger piece shall have as
small a portion as possible cut from it.
Can you cut this piece of linoleum into four pieces that will fit together and form a
perfect square? Of course the cuts may only be made along the lines.
"So various are the tastes of men."