# Reconstructing points based on the sum of their coordinates

9 points are drawn on a piece of paper with the following rules:

• Each point has integer coordinates $$(x,y)$$ that are between 1 and 10 inclusive.
• For each point there is exactly one other point so that their x-coordinates or their y-coordinates match.
• Two points cannot sit on top of each other.

The sum of the coordinates of each point (ie., x+y) is provided: 2, 5, 6, 7, 8, 10, 11, 12, 13. Can you reconstruct the location of each point? Bonus question: can you find multiple solutions? Good luck!

This puzzle was inspired by this one: The Grid World - Catastrophe

The second rule states: "For each point there is exactly one other point so that their x-coordinates or their y-coordinates match". So we should be able to pair each point. We have an odd number of points to place, so

It is impossible.

ps:sorry I can't comment yet.

• Would 1,2 / 1,3 / 2,3 not satisfy that second rule? Aug 11 '20 at 11:11
• @Mohirl The way I undestand the second rule, your second point would not satisfy it (same x as first point and same y as the third). Another puzzle on this theme with a modified 2nd rule has been made: puzzling.stackexchange.com/questions/100996/… Aug 11 '20 at 12:04

I hope I understood correctly.
Solution 1:

1, 1 (2)
1, 4 (5)
2, 4 (6)
2, 5 (7)
3, 5 (8)
3, 7 (10)
4, 7 (11)
4, 8 (12)
5, 8 (13)

Second solution

same as the first one but you reverse x and y coordinates for each point
1, 1 (2)
4, 1 (5)
4, 2 (6)
5, 2 (7)
5, 3 (8)
7, 3 (10)
7, 4 (11)
8, 4 (12)
8, 5 (13)

Explanation:

I started of with the first sum which is 2. This means the first point has the coordinates 1 and 1 since they have to be integers between 1 and 10.
the idea: Since each 2 points share either only x or only y, there are no 3 points on the same line.
So the safest way to continue would be to move to the right with the next point or up alternating from point to point like in the image below made with my awesome paint skills. Then I took all the sums in order and tried to reach the next one by changing X on the first move and Y on the second move.
For the second solution I changed Y on the first move and X on the second one.
So starting at (1, 1), changing X to match the next sum (5), I ended up with (1, 4). Then changed Y to reach the next sum (6) and ended up with (2, 4) and so on.

• This was the intended solution, but I stuffed up the description, because here each point has multiple matching partners. So the actual solution is the other answer. Aug 10 '20 at 22:28
• @DmitryKamenetsky: Surely editing the question to clarify the description would be a valid option? To be honest, when I first read it, I read it the way you indented it to be, and not the way the accepted answer understood it. Aug 11 '20 at 8:29
• By the time I realised my mistake, there were already two answers. So i felt it wouldn’t be fair to edit the question at that stage. Aug 11 '20 at 9:35