This design is made of three 2x1 rectangles. What fraction of it is shaded?pic.twitter.com/7G5Q61eolT
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Mathematical physicist and mentor to mathematically talented youth. Talent is that which bridges the gap between what can be taught and what must be learned.
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This design is made of three 2x1 rectangles. What fraction of it is shaded?pic.twitter.com/7G5Q61eolT
Shaded area is cot(2arctan(1/2)), interestingly enough. Still struggling, though, to see why the answer here is what it is. Seven copies of the shaded portion should somehow reproduce the whole figure. Hmm.
Yes, I thought this was an odd answer too. I was hoping someone would provide some magically clear dissection, but it hasn’t happened yet. Maybe this one is just hard!
Yes. Indeed, it's worth remembering, and acknowledging, that sometimes things really are just hard, all right. But dissections into seven pieces are tantalizing things; when possible, as they sometimes are, they tend to make a strong and lasting impression. I haven't given up.
I have a sneaking suspicion that it may all ultimately trace back to the relations 3 = 2^2 - 1^2 4 = 2*2*1 5 = 2^2 + 1^2 that link the side lengths of the triangles from which the whole figure is built. Interesting, too, that 7 = 2^2 + 2*1 + 1^2. Hmm.
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