# Distance Puzzle

The rules are:

1. Draw two circles (O) in the empty cells in every row and column.
2. The numbers in the black cells indicate the space between the 2 circles in the black cells' row or column, in either order.

3. Black cells also count as spaces.

Puzzle:

Example:

Tabular transcriptions of the puzzle and example

Puzzle:

1,2
4,8
6,6 O
6,0
2,2
7,4
0,2
2,0
4,1
2,7

Example:

O O 1,2
O O 3,5
O 3,1 O
O 2,3 O
O 1,3 O
O 5,0 O
2,3 O O
• So in every row and column exactly two cells need to be marked by an O, and the given numbers indicate the distance between the two O's in that row and that column in either order. – Jaap Scherphuis Mar 4 '19 at 12:08
• Nice puzzle, some ideas for improvements: That the clues can be in either order should be highlighted. You give two examples but both use the first number for rows and the second for columns (In your example all clues are row first, column second). It might help to put the clues in a consistent order, for example always lowest to highest. – w l Mar 4 '19 at 15:43

Here's a solution with a fully worked-out logical path. Blue means "no circle here"

Step 1:

The given circle must have a separation distance of 6 from the other circle in its row (as the black square in its row is (6, 6) so distance of 6 for both row and column).

Step 2:

Either the row or column for the (4, 8) black square must have circles with separation distance 8 - on a 10x10 grid, this mean circles on either edge. If it is the row, then the (1, 2) black square will have separation distance of 0 in its column, a contradiction. Therefore the 8 clue applies to the column of the (4, 8) black square.

Step 3:

For rows/columns with only one circle, color in blue any squares that have a separation distance other than the two allowed distances. (The two allowed distances being the numbers in that row/column's black square)

Step 4:

There is only one way to place circles with separation distance 6 for the (6, 0) black square. Then since the (4, 8) black square already has its 8 clue satisfied, the new circle in its row must have separation distance 4. There is only one way to complete this row now.

Step 5:

There is only one way to place circles 6 apart for the (6, 6) black square's column. We can then trivially complete the (1, 2) black square's clues.

Step 6:

The (7, 4) black square's column only has barely enough space for separation distance 4, and won't fit 7. Using 4 as the column clue and 7 as the row clue allows it to be satisfied.

Step 7:

First trivially completing the row clue for the (6, 0) black square (it must be 0), then notice that there is only one way to fit a separation distance 2 for the (2, 2) black square's row. The (2, 2) black square can be completed.

Step 8/solution:

The rest is trivial deduction.

I think that this might work...