# Alternating series, a crafty solution.

Simon learned this from an alternating series visualization by Think Twice.

# Area of a dodecagon without trigonometry How do you find out the area of this dodecagon without using trig? Rearrange the triangles to make… Three squares! The area of the dodecagon with a radius r is equal to the area of three r-sided squares or 3r^2. The formulas for other polygons. There seem to be no formulas for the heptagon, nonagon and hendecagon (without using trigonometry that is). Simon’s notes above also say that no polygon can possibly have an area equal to or larger than πr^2 (because that’s the area of a circle). A square is 2r^2, a dodecagon is 3r^2, and so no polygon is 4r^2.

Simon learned this from a visual mathematics video by Think Twice.

# Simon’s Formula to Check Triangle Numbers

Simon spent the morning of December 5 pondering about how to test whether a number is a triangle number. “To test if something is a triangle number: double it, ask if it’s a multiple of its own square root. If that square root has a decimal, round it down”. This was his initial hypothesis, later discarded. Simon jotting equations on a piece of cardboard while I drag him to his French lesson as we’re running desperately late. I feel bad about interrupting his train of thought.

Another formula he came up with was if n is even, m is a triangle number. After we got back home, he quickly wrote some code to check it:

# Peg Solitaire

Simon proving his peg solitaire solution:

In a game of peg solitaire, if you win you must end up on one of these 5 points! This analysis was a little more difficult than with the pyramid. Because in the pyramid, I ended up with a symmetric picture whereas with this one I ended up with an asymmetric picture. So I had to do one more step of removing all of the x’s that break the symmetry:

# Divisibility by 3

Nice little trick for divisibility by 3

# 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