Match Sticks in the Summer

Match sticks come in handy during the summer. You can use them for igniting charcoal to get a barbecue going, setting off fireworks, and lighting those citronella candles that keep the bugs away.

They also remind me of a summer 10 years ago when I attended the Eugene Strens Memorial Conference on Recreational Mathematics and Its History (Strens conference, for short). Held at the University of Calgary in Alberta, this gathering of puzzle enthusiasts remains in my mind as the most enjoyable of any math meeting that I have ever attended.

One of the presenters at that meeting was mathematician Heiko Harborth of the Braunschweig Technical University in Germany. "Match sticks are [among] the cheapest and simplest objects for puzzles which can be both challenging and mathematical," he insisted. He handed out a box of matches to anyone who preferred working on match-stick puzzles instead of listening to his lecture.

One group of match-stick problems that he described involves constructing patterns in which a given number of sticks meet end to end, without crossing each other, at every point in a geometric figure on a flat surface. For example, a figure made up of three sticks laid out as an equilateral triangle has two sticks meeting at each corner. Three sticks is the smallest number that can be used to create a pattern in which two sticks meet at every vertex.

The problem is tougher when three sticks must meet at every corner. The answer requires a figure made up of a minimum of 12 sticks that meet, three at a time, at eight vertices.

Patterns in which two match sticks meet at every vertex (left) and three match sticks meet at every vertex (right).

What about four sticks meeting at each vertex? The smallest known construction 10 years ago was Harborth's arrangement of 104 match sticks meeting at 52 points. That pattern appeared on the front cover of Science News two months after the conference, when my article on recreational math was published. To Harborth's knowledge, no one has yet found an arrangement that requires fewer match sticks. It is known, however, that no such patterns exist for five or more sticks meeting at every vertex.

The best known answer to the problem of arranging match sticks so that four sticks meet at each vertex.

Playing with match sticks raises questions that pertain to graph theory - the study of ways in which points can be connected. Graphs often play important roles in circuit and network design, and match sticks provide a way of exploring the characteristics of certain types of graphs.

For example, a line segment is the only possible figure using a match stick to connect two points. With two or three match sticks, there are two configurations connecting three points: The points fall in a line or a triangle. There are five different arrangements that connect four points, thirteen arrangements for connecting five points, and 50 connecting six points. But no one has found a general formula for determining the number of possible arrangements for any given number of points.

As the number of points increases, the number of possible match-stick patterns escalates rapidly. Can you find the 50 patterns that join six points?

One can also try producing all match-stick graphs for which the number of match sticks is given. For 1, 2, 3, 4, and 5 match sticks, the number of different graphs is 1, 1, 3, 5, and 12, respectively. Can you find all 28 graphs that can be made with six match sticks?

Many more examples of match-stick puzzles appear in the paper by Harborth published in The Lighter Side of Mathematics, the proceedings of the Strens conference.

The meeting organizers originally didn't plan to publish a proceedings. Fortunately, good sense prevailed, and the final volume appeared 8 years after that singular summer. Now, anyone can ponder the symmetries of frieze patterns, the 50,613,244,155,051,856 ways to score in a bowling game, group theory applied to Rubik's cube, curiosities of Euclidean geometry, alphamagic squares, conundrums presented by match sticks arranged in the plane, and much, much more.

Clearly, recreational mathematics offers a highly pleasurable entree into more "serious" mathematical pursuits.

References:

Gardner, Martin. 1969. The Unexpected Hanging and Other Mathematical Diversions. New York: Simon and Schuster.

Guy, Richard K., and Robert E. Woodrow, eds. 1994. The Lighter Side of Mathematics: Proceedings of the Eugene Strens Memorial Conference on Recreational Mathematics and Its History. Washington, D.C.: Mathematical Association of America.

Peterson, Ivars. 1990. Islands of Truth: A Mathematical Mystery Cruise. New York: W.H. Freeman.

______. 1986. Games Mathematicians Play. Science News 130(Sept. 20):186-189.

Heiko Harborth can be reached at h.harborth@tu-bs.de.

Comments are welcome. Please send messages to Ivars Peterson at ipeterson@maa.org.