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CronoDAS 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 solved both of them, slowly, in a sleep-deprived state. For the continued fraction, I first tried doing successive approximations to see what the answer "should" be... when I got 1.41 I figured that it was probably the square root of 2. So the next thing I did was to try squaring the expression, which wasn't exactly helpful, but it did lead me to notice that the continued fraction contained itself so I could use the algebra trick that Luke_A_Somers used.