[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.

[Guest]

8

[bonanova]

8

[1]

b2-b5 = d1-d4

b2-d1 = b5-d3

[2]

a1-a5 = b7-f7

a5-b7 = d1-f2

[Guest]

What is the largest number of pawns on a 7x7 chess board whose pairwise distances are all different?

6. It is the theoretical maximum, for which there are several solutions (e.g. A1, A2, C1 C4, D5, G6).

[bonanova]

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?

[Guest]

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.

[Guest]

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

[bonanova]

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.

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.

[Guest]

Nice try but check B5-G7 with D2-F7, and B5-G2 with D2-G7.

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.

Um, right. Forgot the first one adds zero. So there probably is a solution with 7 pawns.

Also, there are only 26 different spacings...

[bonanova]

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.

[Guest]

ugh man this was an annoying problem. i finally broke down and wrote a cpu program to solve it.

A1,A4,C1,C2,F6,G3,G7

[bonanova]

ugh man this was an annoying problem. i finally broke down and wrote a cpu program to solve it.

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.