Geometric Vanishes, A Little History

  

The image above represents one of the most famous "vanish" puzzles of all time. It was created in 1898 by Sam Loyd, America's greatest puzzlist at the end of the 19th century. It is said that during his life time over ten million copies of this puzzle were published around the world. Since then there have been almost that many variations created, including several variations by Sam Loyd himself. Several of these can be seen on Robs Puzzle Page involving cards, eggs, cowboys and more.

The original puzzle had Chinese warriors around the rim of a circular piece of cardboard fastened at the center to a larger piece of cardboard so it can be turned. Part of each warrior is inside the circle and part is outside.
Count carefully, when rotated from its initial position to its second position, one warrior disappears! The challenge is to explain how this "disappearing act" works.

The animation above comes from a web site which also includes a copy you can print and make your own version. (Sorry, it seems to have gone away.)

For students, these may be less instructive than the more obviously geometric vanishes. One of the most popular in classrooms is the Curry Triangle Paradox shown below (Not to be confused with Curry's Paradox in Logic).

The Wikipedia Article includes, "According to Martin Gardner, this particular puzzle was invented by a New York City amateur magician, Paul Curry, in 1953. The principle of a dissection paradox has however been known since the start of the 16th century."

A nice video by James Tanton gives an explanation of the paradox if the reader can't explain it.

I found information about the earliest "vanish" puzzle at a puzzle blog by Marianno Tomatis .
The first example of vanishing area puzzles was reported in the 1566 book Libro d'Architettura Primo by Sebastiano Serlio (1475-1554), an Italian architect of the Renaissance. He cites the case of a 3×10 rectangle which can be converted into a rectangle 4×7 and two triangles 1×3. Curiously, Serlio didn't seem to noticed that the sum of the three resulting areas (31 square feet) was greater than the original one (30 square feet):

In 1769 Edmé Gilles Guyot (1706-1786) described the vanish area paradox in the second volume of his Nouvelles récréations physiques et mathématiques (read it here): a 10 × 3 square can be divided in four pieces which, rearranged, define a 5 × 4 and a 6 × 2 rectangle. 

This puzzle is often quoted as the first vanish puzzle but attributed to William Hooper in his 1774 Rational Recreations.  The fact that Hooper took the puzzle directly from Guyot is made clear when it is pointed out that the original edition of Guyot had the puzzle mis-drawn.  The same mis-drawn image appeared in Hooper's first edition.  Both second editions had the error corrected. 

Hooper's book is a collection of many scientific experiments and amusements, and includes "an Appendix of Miscellaneous Recreations".

Probably the biggest boost to a rebirth of interest in these puzzles came from an article by the great Martin Gardner when he wrote about such geometric vanishes in his Mathematics, Magic and Mystery in 1956. If you are interested in this type of puzzle, this book is still a great resource.

Before I leave this topic, I should point out the first version I ever encountered which I found a nice illustration of at a puzzle page by Dr. Ron Knott from the UK.

  

This is the one I most think can be used effectively in the classroom.   I have seen a 13x13 that converts to an 8x21 that is credited to Fibonacci.  Any Fibonacci square can be converted to a product of the Fibonacci numbers before and after.   The error is easier to spot with smaller areas.  Try this one used by H E Licks in his 1917 Recreations in Mathematics.  

  

Enjoy