I think that last value for 100 from previously was meant to be 834 instead of 832
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I've looked into this a bit more, but for variables other 5. this is what I found:
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I looked at 3 variables. For example "1,2,3" which is the solution for target 2. Every even number has a single solution based on this ratio.
1,2,3
2,4,6
3,6,9
I found no solutions for odd numbers though.
====================================
I looked at 4 variables. For example "1,2,2,3" which is the solution for target 2. The same pattern as for three variables arises with a single solution for every even number, and none for the odd numbers
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The case for 5 variables is stated previously.
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I looked at 6 variables. Here I found a similar pattern to the 5 variable pattern. Here is my description for calculation;

EVENS
((N divided by 2) +1)+(N minus 4)+(N minus 10)+(N minus 16) + ...........for any positive number.

(Jumping by 6 every time as before)

ODDS
(N minus 1) + (N minus 7) + (N minus 13) + ..........for any positive number.
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I looked at the case for 7 variables. Here the pattern changes, here is where it gets much complicated. I haven't produced a method for calculation at 7 variables yet. Here are my initial results followed by my thoughts.

Counting these by hand gets a bit tricky so mistakes could have crept in. Now it seems to relate to powers of 2 maybe.
I found that halving N and squaring the result seemed to relate to the grouping of the sets for each target.

I think that last value for 100 from previously was meant to be 834 instead of 832

===========================

I've looked into this a bit more, but for variables other 5. this is what I found:

====================================

I looked at 3 variables. For example "1,2,3" which is the solution for target 2. Every even number has a single solution based on this ratio.

1,2,3

2,4,6

3,6,9

I found no solutions for odd numbers though.

====================================

I looked at 4 variables. For example "1,2,2,3" which is the solution for target 2. The same pattern as for three variables arises with a single solution for every even number, and none for the odd numbers

====================================

The case for 5 variables is stated previously.

====================================

I looked at 6 variables. Here I found a similar pattern to the 5 variable pattern. Here is my description for calculation;

EVENS

((N divided by 2) +1)+(N minus 4)+(N minus 10)+(N minus 16) + ...........for any positive number.

(Jumping by 6 every time as before)

ODDS

(N minus 1) + (N minus 7) + (N minus 13) + ..........for any positive number.

====================================

I looked at the case for 7 variables. Here the pattern changes, here is where it gets much complicated. I haven't produced a method for calculation at 7 variables yet. Here are my initial results followed by my thoughts.

target 2 : 2 sets

target 3 : 2 sets

target 4 : 7 sets

target 5 : 8 sets

target 6 : 15 sets

target 7 : 16 sets

target 8 : 32 sets

target 9 : 34 sets

target 10 : 60 sets

target 12 : 107 sets

Counting these by hand gets a bit tricky so mistakes could have crept in. Now it seems to relate to powers of 2 maybe.

I found that halving N and squaring the result seemed to relate to the grouping of the sets for each target.

Let me know what you think