# All distances different on a chess board

Here is a simple formulation for, I believe, a quite difficult problem.

I have played with it, I don't have the answer yet.

The question: How many pawns can you put on a standard 8x8 chess board in such a way that the distances between two pawns are all different?

Needless to say, each pawn must be exactly centered on a square of the board.

Computers are allowed. Without it, it is quite laborious to even check the validity of a solution.

If no proof of optimality is given (answering to "how many") then my vote goes to the solution that has the most pawns on the board.

• @Taco I think it is clear $\sqrt 5$
– z100
Nov 19 '21 at 20:31
• @Taco Yes, it is usualy called "Pythagorean theorem" :-)
– z100
Nov 19 '21 at 20:54
• @z100, well they're right though in that the question doesn't say which distance metric would be used. Though using something other than the usual geometric distance would probably make this relatively trivial... Nov 20 '21 at 10:16
• For results in larger grids see this sequence: oeis.org/A193838 Nov 22 '21 at 5:09
• @DmitryKamenetsky Also A271490 Nov 30 '21 at 19:28

The maximum is

7

For this solution, the squared distances are

$$\{1,2,4,5,9,10,13,16,17,26,29,34,37,40,45,49,53,58,65,85,98\}$$

You can solve the problem via integer linear programming as follows. Let binary decision variable $$x_{i,j}$$ indicate whether a pawn is placed on square $$(i,j)$$. For each pair $$(i_1,j_1)$$ and $$(i_2,j_2)$$, let binary decision variable $$y_{i_1,j_1,i_2,j_2}$$ indicate whether $$x_{i_1,j_1} \land x_{i_2,j_2}$$. For each distance $$d$$, let $$P_d$$ be the set of pairs $$(i_1,j_1)$$ and $$(i_2,j_2)$$ such that $$\sqrt{(i_1-i_2)^2+(j_1-j_2)^2}=d$$.

The problem is to maximize $$\sum_{i,j} x_{i,j}$$ subject to \begin{align} x_{i_1,j_1} + x_{i_2,j_2} - 1 &\le y_{i_1,j_1,i_2,j_2} &&\text{for all pairs (i_1,j_1) and (i_2,j_2)} \tag1\\ \sum_{(i_1,j_1,i_2,j_2) \in P_d} y_{i_1,j_1,i_2,j_2} &\le 1 &&\text{for all d} \tag2 \end{align} Constraint $$(1)$$ enforces the logical implication $$x_{i_1,j_1} \land x_{i_2,j_2} \implies y_{i_1,j_1,i_2,j_2}$$. Constraint $$(2)$$ prevents more than one pair per distance.

• Row 2 column 5 is directly next to another tile diagonally, same as row 1 columns 1 and 2. Do diagonals not count, or have I misunderstood the question? Nov 19 '21 at 20:39
• The Euclidean distances are $1$ and $\sqrt{2}$. Nov 19 '21 at 20:41
• Ah, okay, so it's just over my head mathematically lol got it 😁 Nov 19 '21 at 20:42
• You can also place this many pawns on a 7x7 board. In that case, the solution is unique up to rotation/reflection. Nov 19 '21 at 23:09
• @DanielMathias I confirm your result. Nov 19 '21 at 23:36