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ahbwramc comments on Innate Mathematical Ability - Less Wrong
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Well, I rather quickly identified the straightforward algebraic way of solving it.
x = 1 + 1/y
y = 2 + 1/y
yy - 2y -1 = 0
Having reduced it to the quadratic formula and a substitution, and lacking a pen and paper, I did not pursue further at the time. Now I'm curious. Let's add 2 to complete the square...
yy - 2y + 1= 2 = (y-1)(y-1)
y = 1 +/- √2
Since X is 1 less than y, these yield X = +/- √2.
I don't find this obvious, even in retrospect.
I tried for maybe thirty seconds to solve it, but couldn't see anything obvious, so I decided to just truncate the fraction to see if it was close to anything I knew. From that it was clear the answer was root 2, but I still couldn't see how to solve it. Once I got into work though I had another look, and then (maybe because I knew what the answer was and could see that it was simple algebraically) I was able to come up with the above solution.