My friend presented me with a puzzle: Construct a geometric shape with 7 points such that there are 6 straight lines, and each line must pass through 3 points.
My friend came up with a solution, but he wasn’t sure, so he consulted me. Actually there was a “model” answer, but he wanted to find another solution. So we had 2 solutions. I’ll present that model answer here so you can understand the puzzle better.
Points A, B and C form an equilateral triangle. Points D, E and F are the midpoints of lines AB, AC and BC respectively. Point G is the centre of the triangle.
Note: 3 points are collinear if they lie on a straight line.
The 7 points are A to G. The 6 lines are formed by ADB, AEC, BFC, AGF, BGE and CGD.
[I’m not going to go too much into math theories and proofs. Besides, I’m not even sure if I can recall them after losing track for so long…]
The first three lines should be easy to understand. D is the midpoint of AB, so ADB is a straight line. So it is for AEC and BFC.
E is the midpoint of AC, and BE is the perpendicular bisector (remember ABC is an equilateral triangle). And G lies on the line BE, since G is the centre of the triangle. So BGE is a straight line.
Using this reasoning, AGF and CGD are also straight lines.
I know, the explanation I gave isn’t quite backed up with math proofs, more with intuitive reasoning. But I’m sure they are (I just don’t know exactly which theorems to cite…).
So I have 2 favours:
- Send me a more proper explanation of the solution given above
- Or send me another solution (remember I have another one?)
I know there are at least 2 solutions. Just wondering if there are more. I will post my other solution, together with any submissions from you next Monday. Or earlier, depending on the number of submissions. You’re welcome to do both. *smile*
Since this is a geometric construction, a picture together with some explanations is appreciated. You can host the image on a site like Flickr, then add a comment to this post with a link to that image and a brief explanation.
Or you can send me your solution via email to vincent at polymathprogrammer dot com. Let me know if you want to remain anonymous (why?), and I’ll just post your solution only.
Your construction doesn’t have to be exact in proportions. Meaning if there’s a right angle, it doesn’t have to be 90 degrees exactly, so long as it looks like it’s 90 degrees. Just add some explanation to supplement the drawing. Try Paint.NET if you don’t have any image editing software.