Tag Archives: algebra

Impossible, Impossible, Impossible

Welcome to this week’s Math Munch!

The Penrose Triangle is an “impossible figure” – or so claim many reputable mathematics sources.  It’s a triangle made of square beams that all meet a right angles – which does sound pretty impossible.  Penrose polygons features in some of M. C. Escher’s most confounding artwork, like this picture:

But, little do these mathematicians know… you can build your own Penrose Triangle out of paper!  Check out these instructions and confound your friends.

Want more optical illusions?  Check out these awesome ones by scientist Michael Bach.

Mathematicians also seem pretty sure that .99999999…. = 1.  Well, trust Vi Hart to show them what’s-what.  Here’s a video in which she tells us all that, in fact, .99999999999… is NOT 1.

Finally, did you know that 13×7=28?  Well, it does.  And here’s the proof:

Bon appetit!  Oh – and April Fools!

Alphametics, Hyperbolic Crochet, and a Puzzle Contest

Welcome to the first Math Munch of December!


Did you know that SEND + MORE = MONEY?  Or that DOUBLE + DOUBLE + TOIL = TROUBLE?  It does if you replace the letters with the appropriate digits!  These very clever puzzles, where the digits in numbers of addition, subtraction, or multiplication problems are replaced by letters in words, are called alphametics (or sometimes cryptarithms).  Mathematician, software engineer, and writer Mike Keith calls them the “most elegant of puzzles” on his page devoted to some alphametics he’s found and created.  Check out the “doubly-true” alphametics – puzzles where the words are numbers – and Mike’s alphametic poetry.  In this poem, written in what Mike calls “Strict Alphametish,” the last word in each line is the sum of the previous words in that line!  Wow!

Next, take a look at these cool objects!

Purple hyperbolic plane

If you draw a line on a hyperbolic plane and a point not on that line, you can make an infinite number of lines parallel to the first line through the point.

These are models of hyperbolic planes crocheted by Cornell University mathematician and artist Daina Taimina.  A hyperbolic plane is a surface that is kind of like the opposite of a sphere: on a sphere, the surface always curves in towards itself, but on a hyperbolic plane, the surface always curves away from itself.

Before Daina figured out how to crochet a hyperbolic plane, mathematicians had no durable, easy-to-use models of this very important geometric object!  But now, anyone with a little crocheting skill (or a willingness to learn!) can make a hyperbolic plane!  Here are instructions on how to crochet your very own hyperbolic plane, and here’s a link to Daina’s blog.

By the way, our favorite mathematical doodler Vi Hart also makes models of hyperbolic planes out of balloons.

Finally, do you like to play with Rubik’s Cubes, stacking puzzles, or other physical math puzzles?  Think you could make one of your own?   These are some of the entries in the 2011 Nob Yoshigahara Puzzle Design Competition.  Here are the winners!  The designer of the first-place puzzle won this cool trophy!

Bon Appetit!