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My question today is spurred by something I came across in chess called The Crooked Path. The idea is simple enough and essentially comes down to this: if the king wants to move say 6 squares up the board, it can either move those 6 squares up in a straight line, or it can instead move 3 squares diagonally up and left, then 3 squares diagonally up and to the right. Note that these two choices of paths (i.e. the straight line or the two consecutive diagonals) both take the same amount of time for the king, 6 turns.

To me this seems to violate the triangle inequality that says if some triangle has sides of length x, y and z, then we must have x+y > z, i.e. that the sum of two side lengths has to be greater than the length of the third side.

Would I be wrong to assert that the discrete geometry of the chess board is what is responsible for violating the triangle inequality? If it is the discrete geometry that is the cause, does there exist a general principle speaking to this phenomena?

I am not too familiar with the principles and general behaviour of discrete geometry, but this certainly struck me as interesting. So any insight on this behaviour would be much appreciated!

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    $\begingroup$ You are trying to insert the idea "triangle" into the world of chess moves. But as you observed, there is no unique distance associated with an arbitrary chess move. For the king piece, the distance travelled can take on two distinct values of geometric distance. So it is not clear that "count of chess moves" can make a distance in some sense. Maybe it can. $\endgroup$ – 311411 Jul 21 at 21:50
  • $\begingroup$ Also note the standard triangle inequality features a non-strict inequality, in the definition of a metric space. But if you have a proper triangle, then yes the inequality is strict. $\endgroup$ – 311411 Jul 21 at 21:51
  • $\begingroup$ 311411, I think my point was really to abstract away from chess. A NxN discrete grid shows up all over the place in math, and certainly one must often like to consider curves or paths in that grid. As (I believe) my example shows, I believe it is possible that certain closed curves (e.g. triangle) on such a grid would exhibit distinct properties unique to the underlying geometry (e.g. grid). We see this in the case of a curved geometry where all the angles inside a triangle can add up to less or more than 180 degrees. $\endgroup$ – cpollack Jul 21 at 22:19
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    $\begingroup$ The Wikipedia article L^p space may be helpful. For values of $p\ge 1$ the distance satisfies the triangle inequality. However, if $p=1$ the inequality is not strict and so you have $x+y=z$ instead of $x+y>z$. $\endgroup$ – Somos Jul 21 at 23:53
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    $\begingroup$ Ahhh ok everyone. I found an article on something called the “Taxi Cab” metric and it describes just this. I had been thinking of the number of moves the king makes from some starting point as the values the metric takes on. Such a metric on a discrete grid has an equality in its triangle inequality according to the “Taxi Cab Geometry” wiki. Another name for the Taxi Cab metric, as @Somos pointed me on to is the $\ell^{1}$ norm. So I guess the answer to the original question I described above is simply that the metric I’m describing in the problem is the taxi cab metric. $\endgroup$ – cpollack Jul 22 at 2:32
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Thanks to discussion from @Somos and @311411 in the comments I have found the answer to my own question. It was pointed out that my question essentially is meaningless unless I define what metric I'm giving to the chess board geometry to make it into a metric space.

I had been thinking of the number of moves the king makes from some starting point as the values the metric takes on. It turns out this metric is exactly the Taxi Cab metric. It also turns out, as the linked article describes, that the Taxi Cab metric is just the metric induced from the $\ell^{1}$-norm, which is known to satisfy the triangle inequality with equality.

Something I cannot speak too is perhaps the "illusion" that the king in the problem seems to travel a longer distance in the same amount of moves by taking the diagonal route, rather than taking the straight route.

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