Challenge Level

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 five numbers whose average is a whole number... **

How 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 $1$, $2$, $3$, $4$, $5$, $6$ is wiped out.

The mean of what is left is $3.6$

Which number was crossed out?

*With thanks to Don Steward, whose ideas formed the basis of this problem.*

The mean of what is left is $3.6$

Which number was crossed out?

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?