Four distinct integers a, b, c, and d have the property that when added in pairs, the sums 10, 18, 19, 20, 21, and 29 are obtained. What are the four integers in increasing order?

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The solution is very easy.

Of the 4 integers a, b, c, d, there are 6 possible sums:

a + b

a + c

a + d

b + c

b + d

c + d

Assume

a + b = 16

a + c = 19

a + d = 20

b + c = 21

b + d = 22

c + d = 25

Subtract the first equation from the second to get

c - b = 3

Now, add this to the fourth equation to get

2c = 24, so

c = 12

The rest follows easily from this:

Since c=12, a + c = 19 implies a = 7

Since a=7, a + b = 16 implies b = 9

Since a=7, a + d = 20 implies d = 13

Since b=9, b + c = 21 implies c = 12

Therefore the answer will be 7, 9, 12, and 13

Done and done!

the volume is the product of the area of the base, 3x(3x+4), and the height, 3x-1

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