# More Puzzles from Maths Is Fun

In an earlier post, I have mentioned that for many games he programs Simon got his inspiration from the site Maths Is Fun. Perhaps I should add that at our home, Maths Is Fun has become an endless source of fun word problems, too! The problem below has been our favourite this week:

Some of the puzzles Simon likes to recreate with paper and scissors rather than program: A version of Connect 4 but this time with the tables of multiplication! Every player is only allowed to move one paper triangle at a time (the triangles indicate which two numbers one can use to get the next product in the table). The one who colours four products in a row wins. As the game progresses it gets trickier

For the jug puzzle game, Simon has developed a graph plotting the winning strategy (analogous to what he once saw Mathologer do for another game). Double-sided numbers, sort of a two-dimensional cellular automaton. The objective is to get to a state when all the numbers would be one colour. The rule: if a cell changes its colour, its four neighbours (not diagonal) also change colour. There’re also other versions of this puzzle with more difficult initial conditions. A number-guessing game based on binary representation. When he was 8 years old, Simon programmed a similar trick in Processing. He also developed the same sort of trick for base 3 numbers.

Simon and Neva have also especially liked the Tricky Puzzles section (puzzles containing jokes).

# Sums of consecutive numbers While waiting to pick his little sister up from a ballet class, Simon explaining general algebraic formulas to calculate the sums of consecutive numbers. He derives the formulas from drawing the numbers as dots forming certain geometric chapes. consecutive integers consecutive odd integers

# Modular Arithmetic visualized with Wheel Math

Simon learned this method from a MajorPrep video and was completely obsessed about it for a good couple of weeks, challenging everyone in our inner circle to factorize numbers using the wheels. Simon’s proof for the 7 section circle. The remainders lie in the smallest circle (for example, the section where all the numbers are divisible by 7 have a zero in the inside circle, and in the section to the right you can see 1 in the inside i.e. all the numbers in this section mod 7 equal 1) 12 sections 5 sections

# Why the Golden Ratio and not -1/the Golden Ratio?

Take any real number and call it x. Then plug it into the equation f(x) = 1 + 1/x and keep doing it many times in a row, plugging the result back into the equation.

At some point you will see that you arrive at a value that will become stable and not change anymore. And that value will be… φ, the golden ratio!

But this equation also has another answer, -1/φ. If you plug that value into the equation, it will be the same, too. The real magic happens once you have rounded the -1/φ down (or up), i.e. once what you plug into the equation is no longer exactly -1/φ. What happens is that, if you keep going, you will eventually reach… φ as your answer!

Simon saw this interesting fact in a video by 3Blue1Brown and then came up with a proof as to why it happens.

Simon also sketched his proof in GeoGebra: https://www.geogebra.org/classic/zxmqdspb

# Simon having fun solving math puzzles on Twitter.

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!

# Inscribed angle theorem “It reveals itself once you complete the rectangle to find the centre. Because, of course, the diagonal passes through the centre once you inscribe a rectangle inside the circle, because of the symmetry”. Tiling the quadrilaterals Simon has crafted applying the inscribed angle theorem. Tiling the “shapes generated by the inscribed angle theorem” “The theorem says that if you have a circle and just three random points on it, then you draw a path between te first point to the second, to the centre, to the third point and back to the first point”.

# Triangular, Square, Pentagonal, Hexagonal Numbers

I asked Simon to show me how he’d come up with the formulae:

# Number Guessing Game

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

# Chaos Game and the Serpinski Triangle

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.