Mathematics would benefit from a wider recognition of the benefits of these kind of explicit thought processes. When the usual techniques aren't sufficient, I propose conducting Mathmerriments (sorta like science experiments, but more fun!) and writing up the results in a... Mathmerriment Report. This is something that I've designed over my spring break, and I'm excited to try it out with my classes once we return to classes. Enjoy :)
It's time for math to take a cue from its friends in science. They figured out a while ago that 87.5% of intelligence is organization, and they acted on this fact, designing... The Lab Report. The main benefit of a lab report is that it's a recognized way to organize one's thoughts for oneself and others. It forces the author to write to a certain standard of clarity, and gives the scientist something explicit to do when they're feeling stuck. It also serves as an aid to the reader, giving the feeling of familiarity.
Mathematics would benefit from a wider recognition of the benefits of these kind of explicit thought processes. When the usual techniques aren't sufficient, I propose conducting Mathmerriments (sorta like science experiments, but more fun!) and writing up the results in a... Mathmerriment Report. This is something that I've designed over my spring break, and I'm excited to try it out with my classes once we return to classes. Enjoy :)
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Shapes are all around. My 9th grade geometers encountered them in some suprising situations!
I was recently introduced to the game Spot It! at a party with friends. The game consists of a deck of 55 cards. Each card has a variety of symbols on it, e.g. a cat, padlock, pencil, knight, igloo, taxi, spider, etc. The symbols are arranged on the cards so that any two cards have exactly one image in common. When playing, two cards are revealed at a time, and players race to shout out the one picture that both cards have. I really enjoyed playing the game, and it got me thinking about what it would take to create a deck of my own. I also noticed that in all the cards provided, every card has the same number of images. The mathematician in me thought it would be nice to impose some additional symmetry: each image should occur on the same number of cards and any two images should occur on some card together. This led me to define the following variables
Counting the total number of images (with the repeated occurrences counted) on all cards two ways (images per card, or cards per image) yields the equation $Pr=Nz.$ We use the symmetries assumed to count the number of images. Let $x$ be a particular image. For any card $S$, that $x$ appears on, there will be $z-1$ other images on $S$. Since there are $r$ cards that $x$ appears on, and any two images appear on exactly one card together, there must be $P=r(z-1)+1$ images. Next we count the number of cards. Let $S$ be a particular card. For any image $x$ on card $S$, there will be $r-1$ other cards with $x$ on them. Since there are $z$ images on $S$, and any two cards only intersect at one point, there must be $N=z(r-1)+1$ cards. Armed with our three equations
If you've been exposed to projective geometry, this should begin to look very familiar. Viewed in a certain light, Spot It! is a carefully-crafted recreation of a finite projective plane where the images play the role of points and the cards play the role of lines. In fact, if you substitute "image" for "point" and "card" for "line" in the first two projective geometry axioms below, you'll get two of the assumptions above.
If $r=z$ is one more than a prime power, then there is a method to create a plane – read "deck" – that we could play Spot It! with. I crunched the numbers, and when there are eight images per card, you can make a deck of 57 cards - two more than are in the Spot It! game. Fortunately, only the first axiom is important to play the game, so if you happen to lose a couple more cards, you can still have a fun time! If you want to check Spot It! out for yourself you can find it on Blue Orange. |
About Me
I started this blog to share my transformation from math nerd to math nerd who loves to share math with young people. I teach high school in Hanoi, Vietnam. Your comments are always welcome. Archives
May 2021
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