### Pair Sums

Five numbers added together in pairs produce: 0, 2, 4, 4, 6, 8, 9, 11, 13, 15 What are the five numbers?

### Summing Consecutive Numbers

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

### 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.

# Unequal Averages

##### Stage: 3 Challenge Level:

This problem follows on from M, M and M so you may want to look at that first.

Here's an interesting set of five numbers: $$2, 5, 5, 6, 7$$ The mean, mode, median and range are all 5.

Can you find other sets of five positive whole numbers where:

Mean = Median = Mode = Range

Can you find sets of five positive whole numbers that satisfy the following properties?

Mode < Median < Mean

Mode < Mean < Median

Mean < Mode < Median

Mean < Median < Mode

Median < Mode < Mean

Median < Mean < Mode

Not all of these can be satisfied by sets of five numbers! Can you explain why?

Show that some of them can be satisfied with sets of just four numbers.
Show that all of them can be satisfied with sets of six numbers.

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