Take the numbers $1, 2, 3, 4, 5, 6$ and choose one to wipe out.

For example, you might wipe out $5$, leaving you with $1, 2, 3, 4, 6$.

The mean of what is left is $3.2$.

**I wonder whether I can wipe out one number from $1$ to $6$, and leave behind an average which is a whole number... **

What about starting with other sets of numbers from $1$ to $N$, where $N$ is even, **wiping out just one number**, and finding the mean?

**Which numbers can be wiped out, so that the mean of what is left is a whole number?**

**Can you explain why?**

**What happens when $N$ is odd?**

**Here are some puzzling wipeouts you might like to try:**

One of the numbers from $1$ to $15$ is wiped out. The mean of what is left is $7.\dot{7}1428\dot{5}$. Which number was crossed out?

One of the numbers from $1$ to $N$, where $N$ is an unknown number, is wiped out. The mean of what is left is $6.8\dot{3}$. What is $N$, and which number was crossed out?

One of the numbers from $1$ to $N$ is wiped out. The mean of what is left is $25.76$. What is $N$, and which number was crossed out?

## Comments

### A Solution

Mr Ho from Renaissance College sent us a solution on behalf of his Year 8 Enrichment class.

They carefully considered this problem in two parts; when N is odd and when N is even. In each case they have presented a clear explanation of their findings. Well done!

Here is their solution.

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