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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:

Simon’s equations to solve the problem
Simon has developed a system to show the relation between the actual time a and time m that a mirrored clock would show: m = 12 – a
Another clock puzzle from Maths Is Fun
Simon’s solution
solving this during his evening tea

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).

Geometry Joys, Math and Computer Science Everywhere, Math Tricks, Murderous Maths, Notes on everyday life, Simon teaching, Simon's sketch book

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
Math Tricks, Murderous Maths, Simon teaching, Simon's sketch book

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
Coding, Contributing, Geometry Joys, Math Tricks, Murderous Maths, Python, Simon teaching, Simon's Own Code, Simon's sketch book

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

Community Projects, Contributing, Math Riddles, Math Tricks, Milestones, Murderous Maths, Simon teaching, Simon's sketch book

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!

Crafty, Geometry Joys, Math Tricks, Murderous Maths, Simon teaching, Simon's sketch book

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”.
Geometry Joys, Math Tricks, Murderous Maths, Simon teaching, Simon's sketch book

Triangular, Square, Pentagonal, Hexagonal Numbers

Applying one of his favorite materials – checkers – Simon showed me the tricks behind polygonal numbers. The numbers written in pen (above) correspond to the actual triangle number (red rod) and the row number (blue rod).
Square numbers
Pentagonal numbers
And the next pentagonal number
(Centered) Hexagonal numbers
Fragment of the next (centered) hexagonal number
The following morning I saw that Simon came up with these general formulae to construct square, pentagonal and hexagonal numbers using triangle numbers. The n stands for the index of the polygonal number. Later Simon told me that he had made a mistake in his formula for the hexagonal numbers: it should not be the ceiling function of (n-1)/2, but simply n-1, he said.

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

Here is a square number constructed of two triangle numbers (the 5th and the 4th, so the nth and the n-1st)
The working out of the same construction. In the axample above n equals 5, so the 5th square number is indeed 25.
The nth pentagonal number constructed using three triangle numbers: the nth triangle number, and two, n-1st triangle numbers.
The working out of the pentagonal number formula
The nth hexagonal number
The formula for calculating the nth hexagonal number from six n-1st triangle numbers plus 1. (Simon later corrected the (n+1) into (n-1)).
Coding, Math Riddles, Math Tricks, Murderous Maths, Python

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

Codea, Coding, Experiments, Math Tricks, Murderous Maths, Simon's Own Code, Simon's sketch book

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.

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.

How come a dot never happens inside one of the black triangles in the middle? – I asked.
Sometimes you start there, but the next dot (half-way towards one of the corners) is already outside the black triangle, Simon showed. (The screenshot above is of such an occurance. If you look carefully, you will see a dot in the middle).