Renea Mielec MA 341 December 2 1999 Pentominoes

Renea Mielec MA 341 December 2, 1999

Pentominoes • Pentominoes are shapes that use five square blocks stuck together so that each square shares at least one whole side with another square. • Classified as a polyomino - a category which also includes: monominoes dominoes triominoes tetrominoes

Basics of Pentominoes • There are 12 different ways to organize pentominoes. • They are often referred to by the letters they resemble. • There also pentacubes which are three-dimensional shapes formed from 5 cubes that touch each other on at least one side. Graphic from: http: //lonestar. texas. net/~jenicek/pentomin. html

Pentomino History • Polynominoes were brought to life by Solomon W. Golomb, 22 year old graduate student from Harvard University. • He presented the topic at a Harvard Mathematics Club meeting in 1953. In his talk, he emphasized the general shapes formed from N unit squares. • Although Golomb came up with the name, the idea of pentominoes has been around since 1907. Henry Ernest Dudeney published the first pentomino problem in his book, Canterbury Puzzles.

Pentomino History cont. • No major pentomino movement followed until Martin Gardner wrote about them in the journal Scientific American in 1957. • People played with many different polyominoes but found the triominoes and tetrominoes to be slightly boring while the hexaminoes and heptominoes were too diverse. Yet, the 12 pentominoes seemed to be just right. • Later, Golomb wrote Polyonimoes in 1965.

Pentomino Puzzles • Numerous puzzles ask for a rectangle with sides: 3 x 20, 5 x 12, 4 x 15, or 6 x 10. • There are 2339 solutions for a 6 x 10 rectangle, 1010 solutions for a 5 x 12 rectangle, 368 solutions for a 4 x 15 rectangle, and 2 solutions for a 3 x 20 rectangle. • If you choose one of the pentominoes, you can construct a scaled duplicate of that specific pentomino using exactly nine of the remaining pentominoes. Uses: I, L, N, P, T, W, Z, V, Y

Interesting Facts • In the film ‘ 2001, A Space Odyssey’, Arthur C. Clarke had the HAL Computer play pentomino games against astronaut Bowman. Picture from: http: //uk. imdb. com/Title/title-exact? 0062622 • Pentominoes can be arranged in an Escherian fashion. For example, the W turns into F and X. Graphic from: http: //pubweb. acns. nwu. edu/~gbuehler/speech. htm

Guenter Albrecht-Buehler • Dr. Albrecht-Buehler is the Robert Laughlin Rea Professor of Cell Biology at the Northwestern University Medical School in Chicago. • In his spare time, he likes to create pictures using pentominoes. Picture from: http: //pubweb. nwu. edu/~gbuehler/gab. htm

At the Rio Grande, 48 x 32, 3/94 http: //pubweb. nwu. edu/~gbuehler/p 117. htm Battle With the Dragon, 32 x 48, 1/88 http: //pubweb. nwu. edu/~gbuehler/p 44. htm

Rising Storm, 48 x 32, 11/98 http: //pubweb. nwu. edu/~gbuehler/pap 4. htm The Child, 32 x 48, 12/98 http: //pubweb. nwu. edu/~gbuehler/pap 7. htm

References http: //pubweb. acns. nwu. edu/~gbuehler/ (various pages in the website) http: //lonestar. texas. net/~jenicek/pentomin. html http: //forum. swarthmore. edu/tour/pentominoes. html Martin, George E. Polyominoes A Guide to Puzzles and Problems in Tiling. The Mathematical Association of America, 1991.