Lines On A Plane

So, when I posted that dots puzzle from last time (like a month ago, you may remember) I hadn't fully thought out the solution and didn't realise just how simple it could be. Anyway, lets take a look at it now.

The original puzzle was worded to have 10 dots such that they "made 5 lines of 4," which struck me as slightly ambiguous, and my initial solution was just using 8 dots in a row and the 5 lines of 4 were all colinear. The initial intended solution is to place the dots at the vertices of a 5-pointed star, a rather pretty solution and if you don't think about it too much seems like it might be the unique solution.

When I was working on it, the solution I found was to place 4 points down vertically, call them a,b,c,d. Next put 3 horizontal from d, call them e,f,g. Next draw lines from a to e, b to f, and c to g (and set things up so that those three lines don't all intersect at a point). The three intersections of those lines define where to put the other three of our ten points. Not a particularily elegant solution, but its what I did.

To try for the general class of solution first let us consider that we have a solution and we number the lines 1 through 5. Now, count the points as the lines go through them. It must be the case that you counted 20 points, as each of the 5 lines went through 4 points, so on average each point was counted twice. With all the solutions we have seen so far we have each point touched by exactly 2 lines, is it poissible to have one touched by 1 or 3?

If it is possible to have a point that has 3 lines going through it, then let us assume lines 1,2, and 3 go through that point and call it O. Line 1 also goes through a,b,c,O, line 2 also goes through d,e,f,O (def must be distinct from abc or line 1 and line 2 would have two points in common and therefore be the same line) and line 3 goes through g,h,i,O (again distinct from abc and def) This accounts for all ten of our points. Now a fourth line cannot go through two of a,b,c, also cannot go through two of d,e,f, and also cannot go through two of g,h,i, which means it cannot go through four points, a contradiction.

This proves that each point must be met by exactly two of the lines. Actually, the problem is much trivialized if you just try to place the lines first. Place any 5 lines on the plane such that no three of them share a point (if you place them randomly, it will almost certianly be good) and those lines define 10 points (thats 5 choose 2). So those 10 points are where you put your points.

If you wanted to obfuscate this problem a little, it would probably be best to reword it as "Place 10 unique points on the plane such that one could place 5 unique lines on the place with each line passing through 4 points." You could also generalise the problem to N lines and N choose 2 points, but that would probably be contrary to the goal of obfuscation.