One of my favorite math sites on the internet is mathpuzzle. It’s written and curated by recreational mathematician Ed Pegg Jr. About once a month, Ed makes a post that shares a ton of awesome math—interesting tilings, tricky puzzles, results about polyhedra and polyominos, and so much more. Below are some of my favorite finds at mathpuzzles. Go to the site to discover much more to explore!
Shapes that three kinds of polyominoes can tile.
Erich Friedman’s 2012 holiday puzzles
A slideable, flexible hypercube you can hold in your hands! Video below.
Next, have you ever wanted to be a movie star? How about a math movie star? Then there are two math video contests that you should know about. The first is for middle schoolers— the Reel Math Challenge. It’s run by MATHCOUNTS, which has for many years run a middle school problem solving contest. (I competed in it when I was in middle school.) This is only the second year for the Reel Math Challenge, but lots of videos have already been created. You can check them out here.
The second contest is for high schoolers and is called Math-O-Vision. The challenge is to make a video that shows “the way Math fills our world.” Math-O-Vision is sponsored by the Dartmouth College Math Department and the Neukom Institute.
Finally, here’s a fun little applet I found called Make-a-Flake. You can use it to make intricate digital snowflake designs.
Two snowflakes from the Make-a-Flake gallery.
Of course, it’s a lot of fun to make non-virtual snowflakes as well—find a pair of scissor and some paper and go for it! For basic instructions, head over to snowflakes.info. And for some inspiration, check out this Flickr group!
Think fast! How many dots are there in this picture?
This beautiful picture comes to you from Brent Yorgey and Stephen Von Worley. If you counted the dots, you probably didn’t count them one at a time. (And, if you did, can you think of another way to count them?) If you counted them like I did, you noticed that the dots are arranged in rings of five. Then maybe you noticed that the rings of five are themselves arranged in rings of five. And then, finally, you may have noticed that those rings are also arranged in rings of five! How many dots is that? 5x5x5 = 125!
In this blog post, Brent describes how he wrote the computer program that creates these pictures. The program factors numbers into primes. Then, starting with the smallest prime factor, the program arranges dots into regular polygons of the appropriate size with dots (or polygons of dots) at the vertices of the polygon.
Here’s how that works for 90. 90’s prime factorization is 2x3x3x5:
As Brent writes in his post, this counting gets much harder to do with numbers that have large prime factors. For example, here is 183:
From this picture, I can tell that 183 has 3 as a prime factor. But how many times does 3 go into 183? It isn’t immediately clear.
When Stephen saw Brent’s creation, he decided the diagrams would be even more awesome if they danced. And so he created what he calls the Factor Conga. If you only click on one link today, click that one. The Factor Conga is beautiful and totally mesmerizing.
For more factor diagrams, check out this post from the Aperiodical. There’s a link to the factor diagram by Jason Davies that we posted about over the summer.
Next up, a few months ago we posted about the puzzles of Sam Loyd – one of which was a puzzle called Get Off the Earth. In this puzzle, the Earth spins and – impossibly – one of the men seems to vanish. This puzzle is a type of illusion called a geometrical vanish. In a geometrical vanish, an image is chopped into pieces and the pieces are rearranged to make a new image that takes up the same amount of space as the original, but is missing something.
Here’s a video of another geometrical vanish:
No matter the picture, these illusions are baffling for the same reason. Rearranging the pieces of an image shouldn’t change the image’s area. And, yet, in these illusions, that’s exactly what seems to happen.
Check out some of these otherlinks to geometrical vanishes. Print out your own here. And think about this: Are these illusions math – and, if it so, how? I came across geometrical vanishes because a friend asked if I thought the Get Off the Earth puzzle was mathematical. He isn’t convinced. If you have any ideas that you think can convince him either way, leave them in the comments section!
Finally, the Math Munch team’s home, New York City, (and this writer’s other home, New Jersey) was hit by a hurricane this week. The city and surrounding areas are still recovering from the storm. Sandy left millions of people without power and many without homes. One way people have tried to communicate the magnitude of what happened is to make infographics of the data. Making a good infographic requires a blend of mathematics, art, and persuasion. Here some of the most interesting infographics about the storm that I’ve found. Check out how they use size, placement, and color to communicate information and make comparisons.
This infographic from the New York Times shows the number of power outages in the northeast and their locations in different states. The size of the circle indicates the number of people without power. Why would the makers of this infographic choose circles? Why do you think they chose to place them on a map? What do you think of the overlapping?
This is part of an infographic from the Huffington Post that compares hurricanes Sandy and Katrina. Click on the image to see the rest of the infographic. What conclusions can you draw about the hurricanes from the information?
This is a wind map of the country captured at 10:30 in the morning on October 30th, the day hurricane Sandy hit. The infographic was made by scientist-artists Fernanda Viegas and Martin Wattenberg. It shows how wind is flowing around the United States in real-time. Check out their site (click on this image) to see what the wind is doing right now in your part of the country!
To those in places affected by Hurricane Sandy, be safe. To all our readers, bon appetit!
Hurricane Sandy is currently slamming the East coast, but the Math Munch Team is safe and sound, so the math must go on. First up, if you’ve visited our games page lately, you may have noticed a recent addition. Pentago is a 2-player strategy with simple rules and an enticing twist.
Rules: Take turns playing stones. The first person to get 5 in a row wins. (5 is the “pent” part.)
Twist: After you place a stone you must spin one of the 4 blocks. This makes things very interesting.
Why don’t you play a few games before you read on? You can play the computer on their website, play with a friend by email, or download the Pentago iPhone app. But if you’re ready, let’s dig into some Pentago strategy and analysis.
Mindtwister CEO, Monica Lucas
Mindtwister (the company that sells Pentago) put out a free strategy guide that names 4 different kinds of winning lines and rates their relative strengths. The weakest strategy is called Monica’s Five, and it’s named after Mindtwister CEO and Pentago lover, Monica Lucas. You can read our Q&A for more expert game strategies and insights. We also had a chance to speak with Tomas Floden, the inventor of Pentago, so it’s a double Q&A week.
As you play, you start to build your own strategy guide, so let me share three basic rules from mine. I call them the first 3 Pentago Theorems. (A theorem is a proven math fact.)
If you have a move to win, take it!This one is obvious, but you’ll see why I include it.
If your opponent is only missing one stone from a line of 5 you must play there. It seems like you could play somewhere else and spin the line apart, but your opponent can play the stone and spin back! The only exception to this rule is rule 1. If you can win, just do that!
4 in a row, with both ends open will (almost always) win. This is a classic double trap. Either end will finish the winning line, so by rule 2 both must be filled, but this is impossible. The exceptions of course will come when your opponent is able to win right away, so you still have to pay close attention.
Up next, check out the beautiful math art of Geometry Daily.
#288 Fundamental
#132 Eight Squares
#259 Dudeney’s Dissection
#296 Downpour
#236 Nova
#124 Cuboctahedron
#136 Tesseract
#26 Pentaflower
#92 Circular Spring
The site is the playground for the geometrical ideas of Tilman Zitzmann, a German designer and teacher, who’s been creating a new image every day for almost a year now! He also took some time to write about his creative process, so if you’re interested, have a read. Visit the Geometry Daily archives to view all the images.
Finally, an amazing resource – the On-Line Encyclopedia of Integer Sequences. What’s the pattern here? 1, 3, 6, 10, 15, 21, … Any idea? Do you know what the 50th number would be? Well if you type this sequence into the OEIS, it’ll tell you every known sequence that matches. Here’s what you get in this case. These are the “triangular numbers,” also the number of edges in a complete graph. It also tells you formula for the sequence:
a(n) = C(n+1,2) = n(n+1)/2 = 0+1+2+…+n.
If you make n=1, then you get 1. If n=2, then you get 3. If n=5, you get the 5th number, so to get the 50th number in the sequence, we just make n=50 in the formula. n(n+1)/2 becomes 50(50+1)/2 = 1275. Nifty. Who’s got a pattern that needs investigating?
The second contest is for high schoolers and is called Math-O-Vision. The challenge is to make a video that shows “the way Math fills our world.” Math-O-Vision is sponsored by the Dartmouth College Math Department and the Neukom Institute.
Finally, here’s a fun little applet I found called Make-a-Flake. You can use it to make intricate digital snowflake designs.
Math Munch 