No. In fact, Otto's solution does not require any diagonal moves. On the other hand, it probably isn't the optimal solution (in terms of max total distance moved). Let's look at the 2x2 case with 8 people and 7 rounds. the following solution: 15 26 37 48 16 25 38 47 36 45 18 27 35 46 17 28 13 24 57 68 14 23 58 67 12 34 56 78 has exactly 4 people making an orthogonal move each round, with the others staying in place, and nobody moves more than 4 times. I'm not sure that this is optimal; it may be possible to have nobody move more than 3 times - which would mean that everybody moves exactly 3 times. (It is, of course, optimal for the total distance moved by all the participants.) This doesn't prove that you can beat Otto's solution on a larger board, but it does suggest it. Note that, if Otto's solution looks like line dancing, this one looks like square dancing - although square dancers usually only want to dance with the dancers of the opposite sex. Franklin T. Adams-Watters -----Original Message----- From: fred.lunnon@gmail.com On 9/1/06, Otto Smith <otto@olympus.net> wrote:
There is a simple solution that requires that 127 people to take 126 king move steps each. One person remains fixed. ... [etc] I've a feeling that diagonal moves are necessary for a solution; but can't put my finger on the precise reason ... WFL
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