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See picture below:

There exist an infinite plane with infinite number of dots. For sake of argument, let's assume they are 1 inch away from each other.

However, below(on your far left) you can see 3 lines already made. The last line is the yellow one.

What you see on the left, are all combinations of possible moves. Move is defined as structure of lines until you reach an empty dot.

Thus, there are 6 combinations of single line(on top). While on bottom you see 10 combinations. 5 of them moves with 2 lines, and 5 with 3 lines.

Total # of combinations is 16.

So far so good?

Well the basic unsolved question are:

* Does formula exist to calculate total number of combinations(16 in this case) just by looking at the initial graph of 3 lines?

** Does formula exist to show the breakdown of all combinations (6 for single line, 5 for 2 lines, 5 for 3 lines?

*** is 16 the biggest number of combinations that you can make from 3 line starting configuration? For instance: Try to calculate # of combinations (1 lines, 2 lines, 3 lines) from this initial variation:

To not spoil the fun, i will just say there is more than 16. So is this the best solution? How we can prove that this is the best we can do? Obviously we can prove that just by doing all 3 line configurations by hand, but what if we take it to next level? What's the best 4 line configuration and how many combinations it has? How about 5,6.7...(n) ? The tree expands quite rapidly, and also few things need to be explained:

Combinations vs Permutations:

Above, there are 4 lines as starting configuration. In this case, there are 36 combinations OR 41 permutations. The reason is because you can go from A > B > C > D or C > B > A > D. Once again, is the a formula to calculate combinations (36) and Permutation(41) from any starting configuration?

Notation + Final info to consider:

Using above notation, you can notice that each configuration is unique and might have different #s of combinations and or permutations. For example:

Anyone with any input, either mechanical or potentially writing a code to get the answers How many combinations/Permutations for each variation with up to 10 lines as a starting configuation, would be greatly appreciated. If your program can handle bigger starting configurations, that's even better!

Thank You and enjoy!