MAGIC SQUARE

The first trick in this group is one in which squares mysteriously appear and disappear.

Buy some graph paper at a stationery counter and rule off half- or oneinch squares into a large 8-x-8 square as shown in illustration A. Cut this into four pieces as

Illustration A

Your presentation might go something like this. Show the 8-x-8 square and tell your audience, "Here is a magic block with 64 squares. Count them. Eight times eight equals sixty-four. This was purchased from a German mathematical genius who later died without revealing its secret. A square comes and goes, appears and disappears, and no one knows why. We must await the arrival of another Einstein to explain the mystery. In the meantime, let me show you what happens. Watch."

Rearrange the four pieces as shown in B, and let your audience see that you now have an extra, or 65, squares:

Illustration B

Again rearrange the pieces, this time as in C. Now a square disappears. The two larger blocks are each 5 x 6, or 30. This makes twice 30, or 60, plus the three connecting squares for a total of only 63!

These tricks work because the small angles of the triangular pieces are not the same as the small angles of the four-sided pieces. If you examine the original 8-x-8 square in illustration A, you will see that line WX takes exactly five squares to move up two squares, while the line YZ takes a little more than five squares to move up two squares.

Illustration C

If you make the trick out of stiff paper and carefully try to put it together in the rearranged forms, being careful to line up the vertical and horizontal lines, you will see that they really don't fit well—there are spaces between the pieces. So using the flimsier paper will cover up the bad fit. If anyone remarks about this, you can say, "I guess my cuts weren't exactly straight." The second trick in this group involves a disappearing line.

Draw a rectangle on a piece of paper or cardboard, and draw one diagonal. Then draw thirteen lines as in illustration D, being sure that the lines are exactly the same distance apart from each other. Use a ruler. Be sure, also, that the first and last lines just touch the diagonal.

Illustration D

Now cut out the rectangle, and cut it along the diagonal so that you have two triangles. When you slide the triangles as in illustration E, one of the lines disappears!

Illustration E

Tell your audience that since 13 is not a lucky number, you are going to make one vanish so that you end up with a neater dozen lines.

This trick works because, though you always end up with one line less than you started with, each line is just a little bit longer than it was originally.

MAGIC TRIANGLE

You will need paper, pencil, ruler, and scissors.

Tell one of your audience to draw a triangle, using the ruler, and then to cut it out. Mark the corners: 1, 2, and 3. Tell them that this is now a magic triangle. Tear or cut off the three corners and rearrange them as shown to form a straight line. You can repeat this with any triangle. It always forms a straight line.

MAGIC RECTANGLE

You will need paper, pencil, ruler, compass, and scissors.

Tell your audience to draw a square or rectangle or any four-sided figure. Use the ruler. It need not be even, as long as the sides are straight. With compass, draw an arc—a partial circle—on each corner of the rectangle without changing the compass. Cut out the rectangle, then cut off the corners and rearrange them as shown. The arcs will form a perfect circle.

These two tricks work because they follow a basic rule of plane geometry. This is that the three inside angles of a triangle always add up to 180 degrees. 180 degrees is a straight line. If you draw a diagonal across a quadrilateral—a four-sided figure—you form two triangles. So all the inside angles add up to 360 degrees. 360 degrees is a complete circle.

MAGIC ENVELOPE

You will need something with which to draw a circle—a compass or small saucer. You will also need paper, pencil, and an ordinary envelope.

Tell your audience that you have a magic envelope. It is magic because the post office delivered it on time. Tell them it is magic because with it you can divide a circle exactly in half and find the exact center of the circle.

Draw a circle on a piece of paper.

Place the envelope on the circle so that one corner just touches the inside of the circle. Make a mark at the two places where the sides of the envelope cross the circle. Using the envelope as a ruler, connect these two points. This line, called a diameter, divides the circle exactly in half. Do this again with the same circle from a different position. The point where the diameters cross is the center of the circle.

MAGIC HEXAGON

Ask your audience if anyone can draw the figure in the first illustration without retracing or crossing a line.

The way to do it is shown in the second illustration.