Let A(n) = {1,2,3,4, … 8} and B(n) = {1,2,3,4, …. 16} be two sets of consecutive integers with no repetition.(adsbygoogle = window.adsbygoogle || []).push({});

Divide the set of elements B(n) into two subsets C(n) and D(n) each comprising 8 different integers and such that every element of B(n) is used with the condition that 8 equations of the form A(i) + C(i) = D(i) can be simultaneously formed without using any element of A(n), C(n) and D(n) more than once.

There are 48 unique solutions each involving the additional limitation that the numbers 1 and 2 from B(n) are associated with the number 1 from A(n) to form the equation 1+1=2 as one of the 8 forms A(i) + B(i) = C(i). I haven’t counted the total number of possible solutions with out this added limitation.

For added complexity, there is a number B(k) in B which can be put into D(n) as above. Then the following sets can be created A`(n) = A(n) + 9; C`(n) = C(n) + B(k); and D`(n) = D(n) less B(k) + 17 and 18, such that there are nine elements in each set and 9 simultaneous equations of the form A`(i) + C`(i) = D`(i) can be formed without repeating an element of A`, C` or D`. Determine what the value of B(k) must be and give a solution for a set of 9 simultaneous solutions (there is only one solution for B(k) ). To illustrate this problem, the set A(n) could have been the set {1,2,3,4} and B(n) could have been {1,2,3,..8} since the problem works with any multiple of 4 for A(n), e.g

1+1 = 2

2+4 = 6

3+5 = 8

4+3 = 7

Here the number 5 can be added to A(n) and 9,10 added to B(n). In this case the number

B(k) = 7 and the new sets are.

1+1 = 2

2+4 = 6

5+3 = 8

4+5 = 9

3+7 = 10

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# Problem of paired sums from two sets of numbers

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