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The four numbers a < b < c < d can be paired in six different ways.  If each pair has a different sum, and if the four smallest sums are are 1, 2, 3,  and 4, what are all possible values of d?

 May 13, 2020
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The three smallest sums are
a+b, which is 1.
a+c, which is 2.
b+c, which is 3.

The next larger sum is a+d=4

We solve this system of 3 equations in 3 unknowns:

a + b = 1
a  + c = 2
    b + c = 3

and get a=0, b=1, and c=2

Since we know that

a + d = 4

we substitute 0 for a, and get:

0 + d = 4
    d = 4

So the only values possible are a=0, b=1, c=2, and d = 4

Answer: d can only be 4.

Hope this helps, whymenotsmart!

 

 May 13, 2020

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