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Some Sums ala Mini-Soduko

Find six different natural numbers and arrange them in this 2 by 3 grid so that the sum of the numbers in each of the two rows is the same and the sum of the two numbers in each of the three columns is the same.

a b c
d e f
That is, you want a + b + c = d + e + f and a + d = b + e = c + f.

Is there more than one solution?


Source: Mathematics Teacher, April 1986

Hint: Look at the sum a + b + c + d + e + f....It must be a multiple of 6. Why?

Also, think about the "story" of how Gauss quickly computed the sum 1 + 2 + 3 + 4 +...+ 98 + 99 + 100.


Solution Commentary: Play with different combinations of six different numbers that are multiples of 6. Could all be even? Could all be odd? Could half of them be odd? Keep asking questions like this ....and eventually a solution can be constructed.

For example, the solution I discovered while playing was:

2 3 7
6 5 1

Can additional solutions be obtained by adapting this one...say, by adding 1 to each number, etc. If this is called a primitive solution, then are there other primitive solutions that can not be obtained from the one shown?