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A recent question about non-attacking nightriders on a chess board made me wonder about the other famous fairy chess piece from that era, the grasshopper.

This moves along queen lines, but only if it can jump over another unit (the "hurdle") to land on square immediately the other side. A grasshopper can capture an opposing unit on the landing square. The hurdle can be of either colour, and is unaffected.

So how many grasshoppers can be placed on a chessboard, so that none is threatened by another?

Here is my quick offering, but I don't know if it's best possible:

20 grasshoppers

So if you are still a little confused about the movement rule, check that in the above diagram e.g. Gb8 can move to e8 (hopping over Gd8) or e5 (hopping over Gd6). On the other hand Gc5 can move to e7, g5, f2 & c3.

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  • $\begingroup$ A tricky decision: do I award the tick to the incomplete human solution, or the comprehensive computer solution. It turns out that there probably is no super-elegant approach (unlike nightriders) so the computer solution is always going to be necessary to completely solve. But Albert.Lang did reach the right number, so he gets the tick, but please do upvote Tom Sirdegas too, as he deserves it $\endgroup$ Commented Mar 2 at 4:14

2 Answers 2

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I can do:

22.

Diagram:

non-attacking grasshoppers

Same count:

more non-attacking grasshoppers

A third one:

and more

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  • $\begingroup$ Aside: how do you make such figures with chess board? $\endgroup$ Commented Feb 28 at 13:37
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    $\begingroup$ @Pranay lichess.org screen shot $\endgroup$ Commented Feb 28 at 13:54
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Results from an exhaustive computer search (DFS with pruning):

There are 12 unique solutions (ignoring rotations/reflections) with 22 grasshoppers. I drew a box around similar solutions. enter image description here

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  • $\begingroup$ Gasp well done! $\endgroup$ Commented Mar 1 at 17:03
  • $\begingroup$ Please upvote this: it deserves it $\endgroup$ Commented Mar 2 at 4:14

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