### Whole Number Dynamics I

The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.

### Whole Number Dynamics II

This article extends the discussions in "Whole number dynamics I". Continuing the proof that, for all starting points, the Happy Number sequence goes into a loop or homes in on a fixed point.

### Whole Number Dynamics III

In this third of five articles we prove that whatever whole number we start with for the Happy Number sequence we will always end up with some set of numbers being repeated over and over again.

# Odd Stones

##### Stage: 4 Challenge Level:

Good start from Hamish in New Zealand

$2$ - $8$ - $17$ goes to $4$ - $7$ - $16$

and from there to $3$ - $9$ - $15$

Ruth from Manchester High School then looked for the odd one out

$4$ - $9$ - $14$ is the impossible arrangement.

Consider the numbers in each circle in modulo $3$.

Modulo $3$ means the remainder amount when you divide a number by three.

In the first arrangement ($6$ - $9$ - $12$) the modulo $3$ value of each pile is $0$.

On each move you take $1$ from $2$ of the piles and add $2$ to the third so the numbers which were all $0$ in modulo $3$ now all become $2$ in modulo $3$, and after that $1$ in modulo $3$, then finally $0$ again.

After that the cycle just repeats over and over again.

For four of the arrangements the initial numbers are all equal in modulo $3$ and whatever you choose as the next move they will stay equal in modulo $3$.

But $4$ - $9$ -$14$ is $1$ - $0$ - $2$ in modulo $3$ and so cannot turn into any of the other four arrangements or be reached from them.

Thanks Ruth. That way of looking at numbers using their modulo value seems like a powerful perspective.