Coding, Milestones, Murderous Maths, Simon's Own Code, Simon's sketch book

Multiplicative Persistence in Wolfram Mathematica

Simon has tried Matt Parker’s multiplicative persistence challenge on Numberphile: by multiplying all the digits in a large number, looking for the number of steps it takes to bring that large number to a single digit. Are there numbers that require 12 steps (have the multiplicative persistence of 12)?

Simon explaining the project

Simon has worked on this for two days, creating an interface in Wolfram Mathematica. He wrote the code to make the beautiful floral shapes above, they are actually graphs of how many steps three digit numbers take to get to single digit numbers (each ”flower” has the end result at its center).

What about the numbers with many more digits than three? Simon has tried writing code to look for the multiplicative persistence of really large numbers and also came up with some efficiencies, i.e. shortcuts in the search process. He did manage to find the persistence for 2^233 (the persistence was 2):


persistence for 2^233

However after he applied one of his efficiencies to the code to be able to search through many numbers at once, the code didn’t run anymore. You can read Simon’s page about this project and see his code here:

https://www.wolframcloud.com/objects/monajune0/Published/persistence.nb

Simon writes:

277777788888899 is the smallest number with a persistence of 11.
The largest known is 77777733332222222222222222222:

This code works with a few efficiencies:
1. They’ve already checked up to 10^233, so we don’t have to check those again.
2. We can rearrange the digits, and the multiplication will be the same. So we don’t have to check any of the rearrangements of any of the numbers we’ve already checked.
3 
3a. We should never put in a 0 (a digit of the number). Because then you would be multiplying by 0, which would result in 0 in 1 step!
3b. We should also never put in a 5 and an even number. Because, in the next step, the number would be divisible both by 5 and by 2, so it’s also divisible by 10. That would put a 0 in the answer, which we saw we should never do!
3c. With similar reasoning (assuming we want to find the smallest number of the type we want), we’ll see we should never put in:
– Two 5s
– A 5 and a 7
– When we put in a… (- means anything, the order doesn’t matter):
1,- , remove the 1
2,2, put 4 instead
2,3, put 6 instead
2,4, put 8 instead
3,3, put 9 instead
So, we can reduce the search space and time collossaly, with just some logic!

Experiments, Geography, Geometry Joys, Milestones, Murderous Maths, Physics, Simon teaching, Simon's sketch book, Trips

The skyscraper that set things on fire

Inspired by Matt Parker’s video  about the uniquely shaped building at 20 Fenchurch Street in London, Simon was very excited to visit this address. In the video below, made on the pavement in front of the skyscraper, Simon shows the geometric proof (he learned from Matt) of why the building’s shape used to let it set things on fire on extremely sunny days.

dsc_0171882036634.jpg

dsc_0172798100731.jpg

dsc_0173902041493.jpg

dsc_0165695875158.jpg

dsc_01581954130795.jpg

dsc_0162130755088.jpg

dsc_0166307377127.jpg

dsc_01642040135013.jpg

dsc_0202384532852.jpg

Exercise, Geometry Joys, Notes on everyday life, Simon teaching

Football

Wait a minute, is that Simon actually playing football? We never thought we would see that happen. All the credits go to Matt Parker. He is the one who got Simon interested in football, or rather – in the football as a geometrical object. Simon also gave us a lecture on the history of the football, its changing shape and aerodynamics.

dsc_0850-1910313921.jpg

Crafty, Murderous Maths

Dodecoration

Simon has been begging me to do this project together for days (fans of dodecahedrons that we are) and finally we made it – our own origami dodecoration! Simon learned this from a video by Matt Parker on Stand Up Math and looked up a dodecahedron graph on Google. In the video, Simon explains the main principles of how to make a dodecoration. The steps of how to fold your paper squares are below.

 

Make sure you have 30 origami paper squares, in 3 or 5 colours

Fold in two, valley fold down

dsc_0659148089495.jpg

Fold again, so that the cross section looks like the letter “M”

dsc_06611714250272.jpg

dsc_0662836541452.jpg

Make sure the  bottom of the “M” is facing you, then fold like below. Repeat for all the 30 squares.

dsc_06631688386994.jpg

Now it’s time to start inserting the 30 pieces inside one another. The small triangular folds should fit together to make the vertices of the dodecahedron. Every vertex should be made up of three pieces of different colours.

dsc_06641714477868.jpg

The dodecahedron graphs (the plan to follow while building):

Five colours

Three colours

dsc_0677830776739.jpg

Crafty, Murderous Maths

Calculating Pi by weighing circle and square

This experiment has been inspired by Matt Parker and his Stand Up Math channel.

As one of our Pi day activities, Simon attempted to calculate Pi by weighing a circle. In the video, you he first explains why this should work: the area of a circle with the radius r and the area of a square with a side of 2r can be expressed as Pi x r^2 and 4r^2 consecutively. This makes the ratio between the area of such a circle and the area of the square equal to Pi/4. In other words, Pi can be expressed as 4 times that ratio. But since both the circle and the square are made of the same material, their mass will also have the same ratio.

The result Simon got was pretty close, considering the low precision of our kitchen scales. As Simon’s math teacher correctly pointed out, the result would be much more precise if we had one thousand kids make their own circles and squares and weigh them, and then took the average of their outcomes.

Crafty, Experiments, Murderous Maths

Calculating Pi with a Pendulum

This video has been inspired by the wonderful Matt Parker and his video on the Stand Up Math channel:

Yesterday was Pi day and we are still celebrating! Simon experiments with calculating Pi with a physical thing, a pendulum. For the experiment, he cut a cord one fourth of the local gravity value (9.8m/s^2), that is 245 cm. One full swing of the cord makes Pi (measured in seconds)! Simon measures the time the pendulum makes 10 swings and divides that number by 10, to get the average duration of a swing.

The values Simon got were pretty close! The closest he got (not in this video, but later that day) was 3,128 sec., which is exactly the same value that Matt Parker got! What is the chance of that?

The formula is t = 2Pi times square root of l over g (where l is the length of the cord and g the local gravity).

Starring the cute 3Blue1Brown Pi. Here is some extra footage, with the 3Blue1Brown Pi riding the pendulum:

Crafty, Milestones, Murderous Maths, Physics, Simon teaching

Wobbly Circles and the Center of Mass

What shape can roll well, other than a circle (wheel)? Two circles, attached together according to a formula involving a square root of two! Simon made these “wobbly circles” inspired by a Numberphile video where Matt Parker talks about how the ability to roll well (as in a wheel) is caused by the constant height of the center of mass (as opposed to a square wheel, whose center of mass goes zigzagging up and down). Wobbly wheels also have a stable height of their center of mass, hence they roll!

Simon also made a transparent version (with mom’s help):

 

DSC_0150

DSC_0151

DSC_0153

DSC_0155

DSC_0174

 

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):

 

dsc_3541483407704.jpg

More topics Simon learned about from the Numberphile channel included:

The Stern-Brochot Sequence:

Stern-Brochot Sequence 16 Jan 2018

Prime Factors:

Prime Factors 16 Jan 2018

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

Checking Mersenne Primes 16 Jan 2018

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:

10958 Problem 17 Jan 2018

Simon’s notes on the 10958 problem:

dsc_3552884572610.jpg

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):

dsc_3526717765599.jpg

dsc_35251678381008.jpg

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

dsc_35351933176286.jpg

dsc_3539119604192.jpg

dsc_35441164956673.jpg

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

dsc_3500164521074.jpg

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

Square-Sum Problem 3D 17 Jan 2018