Murderous Maths

# Simon’s Triangles Conjecture

Simon came up with what he calls a conjecture about the minimum number of equilateral triangles that fit into a larger equilateral triangle. He has discovered that for equilateral triangles that have a length of 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 the minimum number of equilateral triangles that fit into them is consecutively 4, 6, 4, 8, 4, 10, 4, 6, 4, 12, 4, i.e. a sequence with a repetitive pattern. In the two videos about Simon’s Triangles Conjecture, Simon explains this discovery and presents his proof. He supposes that the pattern continues for even larger triangles, but has proven it up to the side length of 12 so far.

Murderous Maths

# More Numberphile-inspired stuff!

More Numberphile-inspired stuff! Simon has been studying Mersenne Primes (2^n – 1) and their relation to perfect numbers via the Numberphile channel and heard Matt Parker say no one has proved that there are no odd perfect numbers (that perfect numbers are always even). In the video below, Simon tries to prove why all perfect numbers are even. Here is Simon’s proof: When calculating the factors of a perfect number you start at 1 and you keep doubling, but when you reach one above a Mersenne prime, you switch to the Mersenne prime, and then keep doubling again. Once you double 1, you get 2, so 2 is ALWAYS a factor of any perfect number, which makes them all even (by definition, an even number is one divisible by 2):

More topics Simon learned about from the Numberphile channel included:

Prime Factors:

Checking Mersenne Primes using the Lucas-Lehmer Sequence. Simon’s destop could only calculate this far:

The 10958 problem. Natural numbers from 0 to 11111 are written in terms of 1 to 9 in two different ways. The first one in increasing order of 1
to 9, and the second one in decreasing order. This is done by using the operations of addition, multiplication, subtraction, potentiation,
and division. In both the situations there are no missing numbers, except one, i.e., 10958 in the increasing case (Source). The foto below comes from the source paper, not typed by Simon, but is something he studied carefully:

Simon’s notes on the 10958 problem:

The Magic Square (adding up the numbers on the sides, diagonals or corners always results in the number you picked; works for numbers between 21 and 65):

Simon also got his little sis interested in the Magic Suare:

And, of course, the Square-Sum problem, that we’ve already talked about in the previous post.

Simon’s 3D version of the Square-Sum problem:

# Triangle Numbers. Simon’s own code

Now this was pretty amazing! Simon’s new own code, that he so nonchalantly wrote while “having a break” from practicing recursive functions, generates “triangular numbers”.

A triangular number or triangle number counts the objects that can form an equilateral triangle. The nth triangular number is equal to the sum of the n natural numbers from 1 to n. The sequence of triangular numbers is

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406 …

“This particular piece of code works with any sequence”, Simon said:

He tried to build the whole triangle but got stuck.

# Math vs Math

Yesterday Simon and I were doing some primary school math (Dutch 6th grade, Belgian 4th grade, on average two grades above Simon’s biological age). It’s something we regularly do to get Simon used to the Dutch testing books and school approach to math. On top of doing the Dutch school math tasks Simon also has math/geometry lessons with a teacher on secondary school level twice a week, which is more his “zone of proximal development”, in Vygotsky’s terms. And on top of that, Simon uses a lot of math while programming (sequences, arrays and animations all require complicated formulae derived from algebra, geometry or physics).

I have made an interesting observation. The level in the Dutch math schoolbook we use is generally below Simon’s level and does not require deep abstract thinking. And yet while doing tasks from that book and similar worksheets Simon gets distracted easily and often produces an impression of “being slow in math”. I’m not literally forcing him to do the tasks, he quite likes having completed them (yesterday he made 20 in a row), but judging by how he is while making those primary school works an outside observer who doesn’t know Simon would never guess the extent to which this little boy loves algebra and logic, would never foresee the heights Simon can reach when actually challenged with proper material. After we were done with the school math, Simon rushed to the desktop to finish his program in Visual Basic. He was programming a sequence of odd numbers with obstacles on the way. The television his sister was watching in the same room was no longer a problem, he didn’t get distracted for one second. After a while he pulled me over to let me see the formula he used to make sure the computer only picked odd numbers. “The remainder of integer divided by two always has be greater than zero!” – he shouted joyfully. This sort of formulae, which he uses casually, are a notch higher in complexity and much more abstract than the primary school tasks. Simon is incredibly quick in reproducing and explaining them. And he no longer looks bored, tired or like he is slow in math when talking about such matters. He devours higher order, complex material as if it was most delicious food and he was incredibly hungry. And – as both his math teacher and I have noticed – he always tries to find the system behind every algebraic notion, to see it from scratch.

The primary schools tasks we were doing yesterday:

The stuff Simon is working on during his lessons with his math teacher. Here – linear inequalities with variables on both sides and word problems:

Simon checking his answers to inequality problems in his self-made “inequality machine”:

Simon writing in C#:

# Sums of consecutive integers

It’s been great watching Simon turn a word problem into an equation today. He was busy with “Sums of consecutive integers” (Practice finding the nth number in a sequence of consecutive numbers based on the sum) on Khan Academy.

# Simon’s elegant sequence

Simon likes to compile sequences in bath. This one was so beautiful I asked him to write it down

# We started getting Brilliant e-mails!

Signed Simon up for Brilliant, a service that e-mails  you quick math problems every day. It took Simon about 5 minutes to solve this sequence. Longer than myself, I must admit. First he jotted some guesses on a sheet of paper…

But then he yelled: It’s to the power of! 2 to the power of 0 is one, to the power of 1 is 2 – that’s how we get 5, and so on. 2 to the power of 5 is 32. 33 + 32 = 65, the correct answer!

The same evening while having dinner he all of a sudden thought of a second way to solve the same sequence. He didn’t even want to eat his dinner any more as he said he enjoyed continuing the sequence so much. He was very quick at adding hundreds and thousands.

He said he could not wait until we got a new task the next morning!

This is a perfect example of how rows of boring sums are totally not motivating at school and the child is slow and unwilling to solve them, but when doing sums is part of something very exciting – solving a personal difficult and exciting math challenge that you’re being e-mailed every day it’s a whole different story.