Learning Objectives

• Use skills learnt in this session to mathematics problem
• Apply while loop to code
• Create a visualisation from data.

$3x+1$

We will practice a number of the skills we have learnt already to look at the $3x+1$ Problem. There are two rules for this problem:

1. If the number is odd, we multiply by 3 and add 1.
2. If the number is even we halve it.

Task 7a

Let’s create a function which takes an input number and returns a number based on whether it is odd or even. Hint: we can use the mod(number, base) function to find the modulus of a number given the base, for example, the modulus or remainder of an even number in base 2 is 0.

Collatz Conjecture

The conjecture goes that if we apply the two rules to any positive whole number (and repeat for its outputs), we will always get into a 2-1-4 cycle. For example:

10 -> 5 -> 16 - 8 -> 4 -> 2 -> 1 -> 4 ...

Whilst not completely proven, this sequence will work for any number MATLAB can handle.

Task 7b

Create a function that determines the number of steps to reach 1. Check that any positive whole number reduces to 1. As the number of steps is unknown, a while loop could be used.

Task 7c

Using a for loop, find the steps it takes numbers 1 to 500 to reach 1. Plot the results using a line graph with points.

Bonus

Create a subplot with the line plot on the left and a histogram showing the distribution of steps on the right. Use help histogram or search histogram MATLAB for help on this plot.