While in Southern France, Simon really enjoyed solving this puzzle (he originally saw in a Brilliant.org vid). He was so happy with his solution he kept drawing it out on paper and in digital apps, and later shared the puzzle on Twitter. This sparked quite a few reactions from fellow math lovers, encouraged Brilliant to tweet new puzzles and now Brilliant follows Simon on Twitter, how cool is that!
I asked Simon to show me how he’d come up with the formulae:
Simon writes: Made a little game where the computer thinks of a number 1-100, and you try to guess it within 7 takes! Hint: the algorithm is called “Binary Search”. https://repl.it/@simontiger/NumberGuessingGame
You can also play the fullscreen version here: https://numberguessinggame.simontiger.repl.run/
Now also a reversed version, where you think of a number and the computer guesses it: https://repl.it/@simontiger/BinaryNumGuessingGame
Monday morning Simon showed me the Chaos Game: he created three random dots on a sheet of paper (the corners of a triangle) and was throwing dice to determine where all additional dots would appear, always half-way between the previous dot and one of the corners of the triangle.
Very soon, he found it too much work to continue and I though he gave up. Later the same day, however, he suddenly produced the same game in Codea, the points filling in much faster than when he did it manually, yet following exactly the same algorithm. To my surprise, what resulted from this seemingly random scattering of dots was a beautiful Serpinski triangle.
Simon finds the explanation on Brilliant.org incomplete, so he started a discussion about it on the Brilliant community page: https://brilliant.org/discussions/thread/games-of-chance-course-marble-problem/?ref_id=1570424
Here is a fun math trick! Simon and Neva have made a 8 x 8 cm square (with an area of 64 cm²) and cut it into four pieces, turning the square into a puzzle. Using the same four pieces, they built a 5 x 13 cm rectangle. But wait a minute! 5 x 13 equals 65, so the area of the rectangle is one cm² larger than that of the square!
They also made a similar puzzle using bigger pieces. A 13 x 13 = 169 cm² square turned into a 8 x 21 = 168 cm² rectangle! So now the area of the rectangle is one cm² smaller than that of the square! What’s gong on?
You have probably recognized the numbers in this trick: 5, 8, 13, 21… Those are Fibonacci numbers! Simon explains, that with Fibonacci numbers, the effect of the rectangle area being greater or smaller than the square area is alternating. Fibonacci have a converging ration to φ (Phi), but not φ. The pieces only look like they are golden ratio bigger/ smaller. In reality, there is a little gap between the pieces in the first rectangle and a little overlap in the second.
Simon has been inspired by Mathologer to build this.
Simon shares his strategy to win a 2048 game. He has also worked out a general formula of what a maximum tile can be in any grid. For a 4 x 4 grid classic 2048 grid that maximum is 2^17 or 131072!
“It’s a lovely coincidence that there are 17 particles known in the Standard Model of particle physics, and 2^17 is also the maximum value tile in 2048. And so LHC 2048 actually exists!” Simon shouted after we had finished filming. Ten minutes later, walking outside, he calculated that when playing simplest version of 2048, the game of 4 on a 2 x 2 grid, the probability of winning (getting 4) is 19% if you do nothing, 54% if you make one move and 27 % if you make two moves. He also proved that in the game of 4, you win with the maximum of two moves.
2048 offers a lot of opportunities for math fun!
Simon (and Neva as his assistant) experimenting with the topology of a paper strip, filming their (almost magical) tricks on a slow motion camera:
Inspired by Tadashi Tokieda’s geometry and topology tutorials on Numberphile.