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.