@mathforge @CardColm I haven’t got very far yet... 😂 I’m aware that I owe you a thought-process, though, so I might try to live tweet my doodling for a bit.
@mathforge @CardColm It feels as though the right-angles are pretty crucial so I need to find some way to keep those and maybe play with the other lengths. One option would be to label them (maybe 3,4,5 would be fun?) but this feels like a lot of shapes and labels. 
@mathforge @CardColm I could avoid labelling right-angles by introducing some rectangles. Not sure I’d call this an improvement though.
@mathforge @CardColm I tried folding the whole thing in half. This has the advantage that the rectangles overlap, so I could consider a single square there. Disadvantage is that the triangle I’m interested in has disappeared. I think this is a dead end for now.
@mathforge @CardColm I feel like it would be nice if this was set up so that a + c = b. Still not sure where this thought is going, and a bit worried I’m getting further away from the original puzzle.
@mathforge @CardColm But it would mean I could overlap the two squares and get something like this. That looks like a straight line across the bottom but I don’t think it should be.
@mathforge @CardColm I seem to have lost the triangle I’m supposed to be finding the area of. Time to backtrack a bit.
This process is weird, by the way. It feels weirdly uncomfortable to be posting so much rambling nothingness where I’m mainly getting stuck.
This process is weird, by the way. It feels weirdly uncomfortable to be posting so much rambling nothingness where I’m mainly getting stuck.
@mathforge @CardColm Yep, this isn’t making sense. I think I need to leave it for a bit and see if I have any ideas later.
‘Progress’ so far looks like this:
‘Progress’ so far looks like this:
@mathforge @CardColm At least now I have a public record of what I usually do when I’m trying to think up puzzles. I’ll point people here when they ask how I come up with them. This felt a bit like a #trymathslive thread where nothing actually gets solved.
@mathforge @CardColm Wait, I had another thought! I could have drawn semicircles instead of rectangles to indicate the right-angle...
...nope, that didn’t really help either. Still looking way more complicated than the original puzzle, which is not the aim.
...nope, that didn’t really help either. Still looking way more complicated than the original puzzle, which is not the aim.
@mathforge @CardColm I guess I could remove one shape by introducing a regular hexagon to replace 2 equilateral triangles. The angles would still work, and I think the sides would too.
Not sure the diagram looks any better, even with one fewer shape.
Not sure the diagram looks any better, even with one fewer shape.
@mathforge @CardColm Tried to write an email I’ve been meaning to do for days and failed at that too, so back to this...
It occurs to me that a potentially nice way to use the hexagon would be to indicate the right angles as well. Something like this:
It occurs to me that a potentially nice way to use the hexagon would be to indicate the right angles as well. Something like this:
@mathforge @CardColm Drawing the same thing out again, neater, to see if I like it.
It’s not terrible, but there’s still a lot going on here. Doesn’t feel like a finished product.
It’s not terrible, but there’s still a lot going on here. Doesn’t feel like a finished product.
@mathforge @CardColm Different tack. Instead of two right-angles triangles, I could use other angles that add up to 180 degrees. It would still be possible to put these two together to make a single triangle, although not so easy to find it’s area.
@mathforge @CardColm Accidentally drew this one so that it looks like the triangles meet in the centre of the hexagon. Not sure if that’s actually feasible when it’s all drawn to scale. I kind of like the design though, it’s a bit like a swinging bell.
@mathforge @CardColm Had to do a fair bit of trial and error but it does seem to work. Might need to use some software to convince myself properly (or do some more algebra!)
@mathforge @CardColm Is it useful? I can sort of do the reverse of what I’d been planning and use the fact that I know the shaded triangle to find the combined area of the other two triangles. It’ll be 1/6 of the hexagon.
@mathforge @CardColm This is definitely my favourite idea so far, even if it is quite a long way removed from where I started!
@mathforge @CardColm Aaand I’ve just seen the far more obvious route to showing why those areas are the same. I still like this puzzle though. I’ve had one more idea about how it might be made even nicer, but you’ll have to wait till tomorrow morning for the final reveal now 😉
@mathforge @CardColm My overall working on this (can you spot all the individual pictures from the thread?) also featuring the orange version that I decided I didn’t like.
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