Tag Archives: polyominoes

Pentomino Puzzles, Knight’s Tours, and Decimal Maxing

Welcome to this week’s Math Munch!

Have a pentomino tiling problem that’s got you stumped?  Then perhaps the Pentominos Puzzle Solver will be right up your alley! Recently I’ve been thinking a lot about using computer programming and search algorithms to solve mathematical problems, and the Pentomino Puzzle Solver is a great example of the power of coding.  Written by David Eck, a professor of math and computer science at Brandeis University, the solver can find tilings of a variety of shapes.  Watch the application in slow-mo to see how it works; put it into high-gear to see the power of doing mathematics with computers!

Next, here’s a wonderful page about knight’s tours maintained by George Jelliss, a retiree from the UK.  He says on his introductory page, “I have been interested in questions related to the geometry of the knight’s move since the early 1970s.” George has investigated “leapers” or “generalized knights”—pieces that move in other L-shapes than the traditional 2×1—and he even published his own chess puzzle magazine for a number of years.  His webpage includes a great section about the history of knights tours, and I’m a fan of the beautiful catalog of “crosspatch” tours. Great stuff!

Multiplication, addition, division: which gives the biggest result?

Last but not “least”, to the left you’ll find a tiny chunk of a very large table that was constructed and colored by Debra Borkovitz, a math professor at Wheelock College.  Debra describes how, “Students often have poor number sense about multiplication and division with numbers less than one.”  She created an investigation where students decide, for any pair of decimals, which is biggest–multiplying them, adding them, or subtracting them.  For 1.0 and 1.0 the answer is easy–you should add them, so that you get 2.  .5 and 1 is trickier–adding yields 1.5, multiplying gives .5, but dividing 1 by .5 makes 2, since there are two halves in 1. Finding the biggest value possible given some restrictions is called “maximization” in mathematics, and it’s a very popular type of problem with many applications.

This investigation about makes me wonder: what other kinds of tables could I try to make?

Debra mentions that she got the inspiration for this problem from a newsletter put out by the Association of Women in Mathematics.  There’s lots to explore on their website, including an essay contest for middle schoolers, high schoolers, and undergraduates.

I hope you found something here to enjoy.  Bon appetit!

Circles, Geomagic, and Marble Calculators

Welcome to this week’s Math Munch!

We gave you a taste of some of Vi Hart’s math art last week with her balloon creations.  This week, we’re featuring some of Vi’s doodling in math class art – her Apollonian gaskets!  An Apollonian gasket is a fractal made by drawing a big circle, drawing two or three (or more!) smaller circles inside of it so that they fit snugly, and then filling all of the left-over empty space with smaller and smaller circles.  Here’s the video in which Vi tells how she draws Apollonian gaskets with circles and other shapes (and how she makes other awesome things like an infinitely long caravan of camels fading into the distance).  And here are some more Apollonian gaskets made by filling other shapes with circles from Math Freeze.

Next, you may have seen a magic square before, a number puzzle in which you fill a square grid with numbers so that each row, column, and diagonal have the same sum.  (Play with one here.)  But have you ever seen a geomagic square?

Magic squares have been around for thousands of years, but in 2001, Lee Sallows started thinking about them in a new way.  Lee realized that you could think of the numbers in the square as sticks of particular lengths, and the number being added to as an amount of space you were trying to fill with those sticks.  That led him to try to make magic squares out of things like pentominoes and other polyominoes, butterflies,  and many other shapes!  Aren’t they beautiful?

Finally, what do marbles, binary, and wooden levers have in common?  Mathematical artist, designer, and wood-worker Matthias Wandel built a binary adding machine that uses marbles and wooden gates!  Here’s a video demonstrating how it works:

Matthias doesn’t only build calculators.  Here’s a marble elevator and a machine that you can take apart and reassemble to make a new track.

Bon Appetit!