July 15, 2010

Puzzling Lines

Artist Sol LeWitt (1928-2007) often featured geometric and combinatorial themes in his numerous creations (see "LeWitt's Pyramid"). Indeed, many of his earlier artworks were explicitly combinatorial.

In 1973, for example, he composed Straight Lines in Four Directions and All Their Possible Combinations. This study consisted of 15 square etchings, each inscribed with one or more horizontal, vertical, and diagonal lines in different orientations.

When Barry Cipra, a mathematician and writer in Northfield, Minn., first saw this set of etchings, arranged in a grid, he was intrigued by how the eye automatically tried to connect the lines from one square to the next. In the arrangement that he saw, however, none of the horizontal or vertical lines went completely from one side of the grid to the other.

Cipra asked himself if it would be possible to rearrange 16 squares (one of them blank), without rotating any of the squares, so that all horizontal, vertical, or diagonal lines are unbroken within a four-by-four grid. That was the birth of a challenging mathematical puzzle.

Notice that some of the dark lines in the figure (adapted from LeWitt's design) continue from one square to the next. Is it possible to arrange the sixteen squares, keeping them in a four-by-four grid and not rotating any of them, so that all the lines go all the way from one edge of the grid to the other? If so, how many different solutions are possible?

Playing with square pieces cut from stiff cardboard, Cipra quickly discovered that the puzzle has a solution. Indeed, it has three distinct solutions.

Rotating any one of the three distinct solutions through 90 degrees, reflecting it, or performing a combination of these two operations generates another, related solution. So does taking the topmost row (or the leftmost column) and moving it all the way to the bottom (or to the right).

"In other words, each solution could be drawn on the surface of a torus," Cipra notes. He went on to prove that there are no other possibilities by showing that all solutions must have the toroidal property.

Cipra's LeWitt puzzle resembles the famous "15" sliding-tile game. A square tray holds fifteen tiles, with one vacant space. The player slides the pieces around in such a way as to rearrange the initial configuration into the desired one. Using such moves, is it possible to go from an arbitrary arrangement of LeWitt's squares to each of Cipra's solutions?

Cipra hasn't tried that yet. However, "it should be straightforward to check whether the three solutions are related by an even or odd number of pairwise interchanges and also how they relate to the initial configuration," he remarks.

At the same time, Cipra adds, "the LeWitt puzzle is hard enough to solve when you have complete freedom to move the pieces."

Cipra's puzzle also serves as a reminder of the essential playfulness and simplicity of LeWitt's art, where logic, clarity, and beauty interact to evoke visual delight and deep thought.

Original version posted June 12, 2000.
Updated July 15, 2010.

References:

Cipra, B. 2002. The Sol LeWitt puzzle: A problem in 16 squares. In Puzzlers’ Tribute: A Feast for the Mind, D. Wolfe and T. Rodgers, ed. A K Peters.

______. 1999. What’s Happening in the Mathematical Sciences 1998-1999 (Volume 4). American Mathematical Society.