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Unequal Averages

Age 11 to 14
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?

A.   Mode < Median < Mean

B.   Mode < Mean < Median

C.   Mean < Mode < Median

D.   Mean < Median < Mode

E.   Median < Mode < Mean

F.   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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