October 27, 2011

Oval Track Puzzles

The puzzle known as TopSpin consists of 20 circular pieces, numbered 1 to 20, filling and sliding along an oval track. TopSpin was introduced by ThinkFun (formerly Binary Arts) in 1988.

Pieces can be moved around the track in either direction, keeping their order. Or any four consecutive pieces can be maneuvered into reverse order. For example, consecutive pieces labeled 1, 2, 3, and 4, can be reversed into the order 4, 3, 2, 1.

TopSpin allows two types of moves. Numbered tokens can be moved around the track in either direction (left) or the order of four consecutive tokens can be reversed.

Interestingly, this puzzle has no impossible positions. Any possible arrangement (or permutation) of the pieces can be turned into any other arrangement. That would not be true if the puzzle had either 19 or 21 pieces.

This puzzle was the subject of a recent Numberplay blog, titled "From Sledgehammer to Scalpel," on the New York Times website. Pradeep Mutalik introduced the puzzle and noted: "The challenge is to create a way of moving a single unit without affecting the rest."

The puzzle has a long history and has been the subject of both research papers and books. In his article "TopSpin on the Symmetric Group," published in the September 2000 Math Horizons, Curtis D. Bennett illustrated how abstract algebra and group theory can be used to analyze the puzzle and develop a strategy for solving it.

In the book Oval Track and Other Permutation Puzzles—And Just Enough Group Theory to Solve Them (MAA, 2003), John O. Kiltinen describes TopSpin, in permutation group terms, as "a concrete realization of the subgroup of the symmetric group S20 which is generated by the twenty-cycle (1, 2, 3, . . . , 20) and the product (1, 4)(2, 3) of two disjoint transpositions."

"Group-theoretically, the puzzle is simple to describe, but from a practical standpoint, it is nontrivial to solve," Kiltinen writes. "This makes it an excellent object of study for students of abstract algebra, giving them a concrete representation of a nontrivial and fruitful application of the theory of permutation groups."

Kiltinen's book comes with software (on a CD-ROM) to try and to study the puzzle, including variants that could not be realized in plastic. The book is currently available from the MAA bookstore at a special bargain price.

In his Numberplay blog, Mutalik poses the following problem: The puzzle's initial configuration has all the tokens in order, except that 19 and 20 are reversed. He asks: How can you move token 19 into its proper place without affecting the order of the others? Could you have done so if the tokens were arranged linearly and not in a loop?

Finally, Mutalik asks, "Can you analyze the analogous problem where you flip the order of five tokens at a time?"

"At ten I was fascinated by permutation puzzles like the fifteen puzzle," Bennett remarked. "At seventeen, I became enamored of the Rubik's cube, and today I still look for puzzles like these whenever I visit a toy store."

"For me today, however, the beauty of these puzzles is how easily they lead to deeper mathematics," he added.

Bennett's article was reprinted in the book The Edge of the Universe: Celebrating Ten Years of Math Horizons (MAA, 2006), edited by Deanna Haunsperger and Stephen Kennedy.

Further References:

Kaufmann, S. 2011. A mathematical analysis of the generalized oval track puzzle. Rose-Hulman Undergraduate Mathematics Journal 12(Spring):70-90.

Wilson, J.H. 1993. Permutation puzzles. College Mathematics Journal 24(March):163-165.

October 11, 2011

Split Strips

"Möbius bands (or strips) are beautiful as objects of art, and their mysterious qualities fascinate those who discover or encounter them," sculptor Larry Frazier wrote in an article titled "Möbius strips of wood and alabaster," published in the Journal of Mathematics and the Arts.

"I'm not a mathematician, but as a sculptor, I have been fascinated by the myriad forms that a Möbius band can take, especially when interpreted in a beautiful piece of wood or stone," he continued.

In recent years, Frazier has often brought his gracefully carved artworks to the annual Joint Mathematics Meetings, where he displays and sells them.

Particularly intriguing examples arise from slicing a Möbius strip along its length. Slicing it lengthwise down the middle produces a single, longer band. Slicing a Möbius strip about a third of the way in from its edge produces two linked bands. In effect you've sliced off its edge to produce a narrower version of the original band. The outer piece has two half-twists, and the inner piece has one half-twist, just as the original, uncut band did.

Doubleslice, by Larry Frazier.

Frazier performs this trick in wood. Unlike paper strips, the two wood components can be easily reassembled into the original Möbius band.

Larry Frazier displays his split Möbius strip.

"It's especially nice in wood, because you can put the pieces back together, and you can see the Möbius it came from," Frazier said. "Making the same two cuts in a paper Möbius just produces two loops of paper, and you can't see the original Möbius they came from."

A reassembled sliced Möbius.

Sculptor Keizo Ushio performs similar magic in split stone (see "Sculpting with a Twist").


Frazier, L., and D. Schattschneider. 2008. Möbius strips of wood and alabaster. Journal of Mathematics and the Arts, 2(No. 3):3, 107-122.

Peterson, I. 1991. Fragments of Infinity: A Kaleidoscope of Math and Art. Wiley.

Photos by I. Peterson

October 7, 2011

LeWitt's Complex Form

To bring a trace of irregularity to his pristine geometric structures, conceptual and minimalist artist Sol LeWitt (1928-2007) started with a polygon sketched on a flat surface. He placed dots at various positions within the polygon. These points were then elevated to different heights, dictating the edges of the resulting three-dimensional, faceted object that eschewed the right angles typical of his earlier constructions (see, for example, "Incomplete Open Cubes").

Constructed from painted aluminum and titled Complex Form 6, this LeWitt sculpture was on display in New York City's City Hall Park as part of the "Sol LeWitt: Structures, 1965-2006" outdoor exhibition.

Photos by I. Peterson