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 numbers in this list is 20. How much greater is the sum of the 3 greatest numbers in the list than the sum of the 3 smallest numbers in the list?

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".

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