- Thread starter ChrisMB
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I completely with TKH, ChrisMB, your description is far to vague. It seems as though you mean to have an \(\displaystyle 8\times 8\) grid or lattice upon which one can move horizontally or vertically. But your grid is made from the midpoints of the squares on a chess board. To move from the upper left grid point to the lower right grid point takes eight horizontal moves to the right and eight vertical moves down. We can arrange \(\displaystyle 8\text{-}H's~\& ~8\text{-}V's\) in \(\displaystyle \frac{16!}{(8!)^2}\) ways. BUT each of those paths is the same length.

You have posted a question on a forum that has professional mathematicians some of whom have contributed to work on graph theory. We think in those terms. Your saying "length of each line necessary to connect the center of all squares to one point on the board " makes little sense without your defining all your terms.

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One way to communicate clearly about what you are asking will be to give some examples. Show us in a drawing what lines you are asking about; and perhaps try to find the answer to the same question in a smaller case, such as 2x2 or 4x4, so we can see better what the answer should look like. (Don't worry about talking to professionals in their language; just pay attention to basic clarity.)

One issue is whether "each" and "all" are referring to one point at a time, or one path going through all the centers, or something else. But the most confusing bit is saying there can be no diagonal lines, but then referring to the Pythagorean theorem.

Finally, how would you see recursion being relevant?