Exercise of the Week #5
Today we’re looking at a comprehension question - the variety that’ll make a student shudder!
As ever, answers will be posted next week. Good luck.
Last Week’s Answers
You’ve probably seen this (or a similar problem) on your travels. It’s one of those where you might get a different answer each time you look at it, so is really an exercise is being systematic. This means employing a strategy to ensure you don’t miss anything.
So, let’s start with the smallest triangles.
As we can see, there are sixteen of those.
Don’t forget to include the triangles which are vertical - they still count.
Next up, we need to look at the medium-sized triangles: each one is comprised of four smaller triangles…
So now we’re up to 20. But wait, there are more of those medium-sized triangles! Don’t forget that they can overlap.
And now we’ve got 23. There aren’t any other medium-sized triangles, so now we’re onto the large triangles. These are made up of nine smaller triangles, and there are three of them.
So that’s it; surely we’ve got them all?
Not quite! There’s one more we mustn’t forget:
That’s right, the entire diagram is itself a triangle, which brings our final total to 27 triangles.
If you counted the little fellow on the top right as a triangle too, that’s fine! Technically he’s a square-based pyramid, or perhaps a triangle-based pyramid, so either way, that’s another four triangles there… eep.
If you’d like a more in-depth explanation of this problem, as well as a complicated formula for working out the number of triangles in a similar diagram of any size, YouTuber ‘MindYourDecisions’ has an excellent video solution.
