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Grandpa's Rock Cakes

Weekly Problem 34 - 2012
Grandpa made a super-heavy rock cake. Five people have guessed the weight of the cake. Can you work out the weight of his cake?

About Average

Age 11 to 14 Challenge Level:

Can you find sets of positive integers that satisfy the following?

1. Three numbers with mean $3$ and mode $2$

2. Three numbers with mean $7$ and mode $10$

3. Three numbers with mean $8$, median $10$ and range $8$

4. Four numbers with mean $7.5$, mode $6$ and median $7$

5. Four numbers with mean $6$, median $6.5$ and range $11$

6. Five numbers with mean $4$, mode $3$ and range $9$

7. Five numbers with mean $4$, mode $2$ and range $6$ (two possible solutions)

8. Five numbers with mean $7$, mode $7$ and range $10$ (three possible solutions)


Extension

Can you find a set of four numbers with mean $4$, mode $1$, median $2$ and range $10$?

How about a set of five numbers?
Or six numbers?
Or...
Or 100 numbers? 


If you enjoyed this problem, you may also be interested in M, M and M.

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