Walkabout
Are all of the numbers in a certain list of 15 numbers equal?
(1) The sum of all the numbers in the list is 60.
(2) The sum of any 3 numbers in the list is 12.
Target question: Are all 15 numbers equal?Statement 1: The sum of all the numbers in the list is 60.There are several possible scenarios that satisfy this statement. Here are two.
Case a: numbers are: {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, in which case
all of the numbers are equalCase b: numbers are: {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 7}, in which case
all of the numbers are not equalStatement 1 is NOT SUFFICIENT
Statement 2: The sum of any 3 numbers in the list is 12.This is a very powerful statement, because it tells us that all of the numbers in the set are equal.
Let's let a,b,c and d be four of the 15 numbers in the set.
We know that a + b + c = 12
Notice that if I replace ANY of these three values (a,b or c) with d, the sum must still be 12.
This tells us that a, b and c must all equal d.
I can use a similar approach to show that e, f and g must also equal d.
In fact, I can show that ALL of the numbers in the set must equal d, which means
all of the numbers in the set must be equal. Since we can answer the
target question with certainty, statement 2 is SUFFICIENT
Answer:
Cheers,
Brent