The average of 7 numbers in a certain list is 12. The average of the 4 smallest numbers in this list is 8, while the average of the 4 greatest...

Denote the numbers as `a_1 lt= a_2 lt= a_3 lt= a_4 lt= a_5 lt= a_6 lt= a_7.`   It is given that:

`(a_1 + a_2 + a_3 + a_4 + a_5 + a_6 + a_7)/7 = 12,`

`(a_1 + a_2 + a_3 + a_4)/4 = 8,`

`(a_4 + a_5 + a_6 + a_7)/4 = 20.`

From these equation we can find the sum of the `3` greatest numbers and the sum of the `3` smallest numbers:

`a_1 + a_2 + a_3 =(a_1 + a_2 + a_3 + a_4 + a_5 + a_6 + a_7)`

`-(a_4 + a_5 + a_6 + a_7) = 12*7 - 20*4 = 4,`

`a_5 + a_6 + a_7 =(a_1 + a_2 + a_3 + a_4 + a_5 + a_6 + a_7)`

`-(a_1 + a_2 + a_3 + a_4) = 12*7 - 8*4 = 52.`

The difference in question is 52 - 4 = 48.

But actually these conditions are contradictory. It is simple to find `a_4,` it is

`(a_4 + a_5 + a_6 + a_7) - (a_5+a_6+a_7) = 80 - 52 = 28.`

But all next numbers, `a_5,` `a_6` and `a_7,` must be at least `28,` therefore its sum is at least `84,` not `52.` So the correct answer is "this is impossible".