bonanova Posted July 11, 2009 Report Share Posted July 11, 2009 What is the largest number of pawns on a 7x7 chess board whose pairwise distances are all different? Assume that the pawns lie in the center of their individual squares, and distances are measured on a straight line joining their centers. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted July 11, 2009 Report Share Posted July 11, 2009 Reveal hidden contents 8 Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted July 11, 2009 Author Report Share Posted July 11, 2009 On 7/11/2009 at 2:58 PM, DeeGee said: Reveal hidden contents 8 Reveal hidden contents [1] b2-b5 = d1-d4 b2-d1 = b5-d3 [2] a1-a5 = b7-f7 a5-b7 = d1-f2 Quote Link to comment Share on other sites More sharing options...
0 Guest Posted July 11, 2009 Report Share Posted July 11, 2009 On 7/11/2009 at 9:24 AM, bonanova said: What is the largest number of pawns on a 7x7 chess board whose pairwise distances are all different? Reveal hidden contents 6. It is the theoretical maximum, for which there are several solutions (e.g. A1, A2, C1 C4, D5, G6). Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted July 11, 2009 Author Report Share Posted July 11, 2009 On 7/11/2009 at 8:00 PM, d3k3 said: Reveal hidden contents 6. It is the theoretical maximum, for which there are several solutions (e.g. A1, A2, C1 C4, D5, G6). This solution might take 2nd place. Whose theory limits it to that number? Quote Link to comment Share on other sites More sharing options...
0 Guest Posted July 12, 2009 Report Share Posted July 12, 2009 On 7/11/2009 at 10:14 PM, bonanova said: This solution might take 2nd place. Whose theory limits it to that number? [spoiler=:shrug: I have no clue. It's mine now, I guess ...]I just enumerated the possible distances (26). When you add pawn i you need to use up i-1 of them. 6 is the largest number for which n + n-1 + n-2 ... + 1 is less than 26. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted July 12, 2009 Report Share Posted July 12, 2009 (edited) Reveal hidden contents it seems to me that there are precisely 7 +6 +5 ... distances. meaning you can just barley do it. here's one i found. A1, B5, D2, E1, F7, G2, G7 i haven't completely checked it but it looks like it works. Edited July 12, 2009 by phillip1882 Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted July 12, 2009 Author Report Share Posted July 12, 2009 On 7/12/2009 at 5:45 AM, phillip1882 said: Reveal hidden contents it seems to me that there are precisely 7 +6 +5 ... distances. meaning you can just barley do it. here's one i found. A1, B5, D2, E1, F7, G2, G7 i haven't completely checked it but it looks like it works. Nice try but check B5-G7 with D2-F7, and B5-G2 with D2-G7. Reveal hidden contents The board provides 27 unique square-square distances: delta row delta column [or v.v.] 1-0 1-1 2-0 2-1 2-2 3-0 3-1 3-2 3-3 4-0 4-1 4-2 4-3 4-4 5-0 5-1 5-2 5-3 5-4 5-5 6-0 6-1 6-2 6-3 6-4 6-5 6-6 Seven pawns have 21 pawn-pawn spacings. 1-2 1-3 2-3 1-4 2-4 3-4 1-5 2-5 3-5 4-5 1-6 2-6 3-6 4-6 5-6 1-7 2-7 3-7 4-7 5-7 6-7 Eight pawns have 28 pawn-pawn spacings - not possible. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted July 12, 2009 Report Share Posted July 12, 2009 On 7/12/2009 at 7:20 AM, bonanova said: Nice try but check B5-G7 with D2-F7, and B5-G2 with D2-G7. Reveal hidden contents The board provides 27 unique square-square distances: delta row delta column [or v.v.] 1-0 1-1 2-0 2-1 2-2 3-0 3-1 3-2 3-3 4-0 4-1 4-2 4-3 4-4 5-0 5-1 5-2 5-3 5-4 5-5 6-0 6-1 6-2 6-3 6-4 6-5 6-6 Seven pawns have 21 pawn-pawn spacings. 1-2 1-3 2-3 1-4 2-4 3-4 1-5 2-5 3-5 4-5 1-6 2-6 3-6 4-6 5-6 1-7 2-7 3-7 4-7 5-7 6-7 Eight pawns have 28 pawn-pawn spacings - not possible. Reveal hidden contents Um, right. Forgot the first one adds zero. So there probably is a solution with 7 pawns. Also, there are only 26 different spacings... Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted July 14, 2009 Author Report Share Posted July 14, 2009 On 7/12/2009 at 2:27 PM, d3k3 said: Reveal hidden contents Um, right. Forgot the first one adds zero. So there probably is a solution with 7 pawns. Also, there are only 26 different spacings... Good point ... I hadn't noticed. And yes there is, but it's not trivial to find. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted July 16, 2009 Report Share Posted July 16, 2009 ugh man this was an annoying problem. i finally broke down and wrote a cpu program to solve it. Reveal hidden contents A1,A4,C1,C2,F6,G3,G7 Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted July 16, 2009 Author Report Share Posted July 16, 2009 On 7/16/2009 at 6:58 AM, phillip1882 said: ugh man this was an annoying problem. i finally broke down and wrote a cpu program to solve it. Reveal hidden contents A1,A4,C1,C2,F6,G3,G7 That's it. There's no way other than luck or a computer to find this solution, and for that reason it's not really a Brain Teaser. I posted it, because it seems that an incremental search is possible using a checkerboard and 7 markers. But each move changes so many [6 distances] things, it's not practical. A while back, I used simulated annealing [i was a colleague of S. Kirkpatrick] to place 106-plus circuits on a chip. That approach would work well here. Kudos for pursuing to the answer. Quote Link to comment Share on other sites More sharing options...
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bonanova
What is the largest number of pawns on a 7x7 chess board whose pairwise distances are all different?
Assume that the pawns lie in the center of their individual squares,
and distances are measured on a straight line joining their centers.
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