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Big Powers

Three people chose this as a favourite problem. It is the sort of problem that needs thinking time - but once the connection is made it gives access to many similar ideas.

Wipeout

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2


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





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