I am 35 years old.
My friend is 53 years old.
35 and 53 are mirror birthdays.
We also looked at this puzzle in binary!
Inspired by this fantastic numberphile video with Ben Sparks we are explored sequences of aliquot sums. An aliquot sum is the sum of proper factors of numbers.
The number 15 has factors 1, 3, 5, 15 but the proper factors don’t include the number itself so 15 has proper factors 1, 3, 5. Adding these together gives an aliqout sum of 1+3+5 = 8.
Now we find the aliquot sum of 8, which is 1+2+4 = 7. Then find the aliquot sum of 7, which is just 1 as 7 is a prime number. If we get to 1 we stop.
So 15 – 8 – 7 – 1.
Try with different starting numbers, do all numbers go to 1?
Aliquot 1 – Program to find the aliquot sum of a number
Aliquot 2 – Program to find the aliquot sequence of a starting number
Aliquot 1
Use the computer to find the sum of the factors of 220.
What is the sum of the factors of the answer?
Which one of the following pairs of numbers behaves in the same way?
607, 917
624, 1112
1184, 1210
1554, 2094
Aliquot 2
Can you find a sequence that goes up and then down?
Can you find a sequence that goes up and then down multiple times? Try 138 …
The number 276 is special as it is the smallest number that we don’t know whether or not it ends up at 1. This is an unsolved mathematics problem.
Today we worked on challenging fraction questions sticking to ancient Egyptian methods – only unit fractions and you can’t repeat a fraction.
For example, can the following be expressed as a sum of distinct unit fractions?
This led to the greedy algorithm and some programming on excel:
See here for all the details and solutions.
We are counting in whole numbers always starting at the number 1.
1,2,3,4, …
Schur numbers tell you the highest number that you can count to using k different colours before you’re forced to have an all same-coloured solution to a + b = c.
Example
Is this a valid colouring for k = 3 (3 colours)?
No, because 2 + 2 = 4, 1 + 5 = 6 and 1 + 6 = 7 and same coloured sums are not allowed.
Challenge
For 1 colour, let’s say red, we can only count up to the number 1 like this:
1
For 2 colours, let’s say red and blue, we can count up to 4 like this:
1 2 3 4
Can you explain why we can’t add the number 5?
What is the highest number you can count to using 3 colours?
You can check solutions here:
Here is a python program that generates all the solutions:
https://www.programiz.com/online-compiler/2S5O1C7J96Ys7
See here for the printable puzzle with further explanations and here for a video including the solutions (that are known so far …).
University of Southampton are running a National Cipher Challenge.
Introduction to some of the ciphers that will be used is here.
Competition is https://www.cipherchallenge.org/
There is a great set of tools to use:
https://www.cipherchallenge.org/tools/
Please sign up for the challenge!
Contact Mrs Fleming on [email protected] if you need the monitoring pin to complete the registration.
Here is a series of puzzles with red and blue hats that will challenge your logical skills.
Today we did one of our favourite puzzles inspired by this Numberphile video by Neil Sloane.
Starting with 2 ones, what is the maximum number you can place on an infinite grid according to the rule:
“You can place a number if all the numbers immediately around it add up to itself”
Here is a way to get up to 4:
But then we are stuck as there is nowhere to put the 5. Can you do better than 4?
You can choose where to place the 2 ones to start with, for example:
This puzzle can be tried with any number of 1s to start with.
What is the maximum number you can place starting with three 1s?
Some solutions are here.
This investigation is based on the ancient Greek Apollonius Problem: Can you draw a circle that just touches three given circles? How many can you draw?
We tried this by hand and then on Geogebra following these instructions.
The maximum number of circles possible was unsolved for more than 2000 years, but we solved it! For the solutions and more see the links below:
