The latest edition of MT contains this puzzle (followed by an excellent article about it by @AlfColes):
This post is an account of what happened when I tried to solve it. I have written it because some interesting things happened while solving it. There are spoilers; I would recommend having a go at the problem before reading on.
I felt as though I had few tools with which to work. I started by drawing lots of diagrams, to try to get a handle on the problem. I was finding it difficult to work out a way into solving the problem.
Some associations were coming to mind, ideas about graphs, block designs, and a vague recollection of Desargues' and Pappus' theorems, although I couldn't remember what these theorems were. It is certainly the case that I was at times fixated on configurations that resembled what I could vaguely recall about these theorems.
Alf's article in MT260 (which is more generally about Projective Geometry) suggests that one should try to solve the problem before reading it. After I had worked on it for an hour or so last night, I sent him an email 'thanking' him for the double frustration of (a) the problem, and (b) not being able to read his article before solving it! I think he must have misunderstood slightly, and replied: "Was your solution projective?"
I worked on the problem for another half an hour or so with this unintended 'hint' in mind (see images 5 and 6 above), and then went to bed having not solved the problem.
I came to the realisation that a set of 7 points (trees) with three lines through them could be created by drawing rays through the vertices of two 'corresponding' triangles, meeting at a sort of vanishing point somewhere, as in a (centre of) enlargement - see the first diagram below. But I could not see how/where the other three trees would fit in with this. I drew the following diagrams in a notebook that was at hand:
Whilst drawing the middle diagram, I realised that I could create other vanishing points if the corresponding edges of the triangles were not parallel to each other - a sort of 'wonky enlargement'. This felt significant, as making all three edges of the triangles non-parallel would create three new vanishing points, which would give ten points (trees). The third diagram is a first attempt to create such a configuration. I felt the need for a more accurate sketch:
This sketch was the watershed moment: it led to the conjecture that the three new vanishing points might be collinear, which would create a tenth line, each line having three points/trees on it. If this was always the case - wishful thinking, perhaps - the problem would be solved! I tried a few more sketches to corroborate my conjecture, and whilst I felt these three new points must be collinear, my sketches were inconclusive:
Next to the second sketch above, you can see that I have made the note 'start with horizon', meaning start the construction with the three possibly-collinear points. Here are the next two sketches in which I tried this:
I realised that some of the points in the construction were fixed by the position of other points, most notably the third of the possibly-collinear points. I found great pleasure in constructing these designs. All of this made me feel that I must be on to something. I started to see the problem as a perspective drawing in 3-d, drawing an eye on the bottom image.
I was pretty sure that the three other vanishing points were collinear, even though my sketches were not quite accurate enough, before constructing it on geogebra (you can play with it here):
I gained a lot of pleasure in working on, and solving, this problem. I am always pleased and strangely surprised with myself when I solve a problem that I have found difficult. Sometimes I give up, often only working on a problem for a longer period of time if it comes from a 'trusted source', if I suspect there may be something valuable to gain from doing it. The unintended hint from Alf - the word 'projective' - was enough to focus my thinking in a productive direction. I witnessed the value of wishful thinking (regarding the collinearity of the three new vanishing points); I might have given up had the final configuration not worked.
It is difficult to continue working a problem if there is nothing to hold onto. I drew lots (!) of diagrams, which performed a number of roles (getting a sense of the problem, forming and testing conjectures). My limited prior experience of Projective Geometry was, I think, largely obstructive. I was trapped in trying certain configurations, and managed only to really snap out of them on waking up next morning. I have no doubt that I was working on the problem in my sleep, although I had absolutely no intention or recollection of doing so.