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JonahSinick 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.
The core of the solution is recognizing that it can be reduced to a pair of algebraic equations rather than finishing off the computations. I was referring to the former in saying "could see how to answer it immediately." An extremely gifted child might also be able to solve the equations without pencil and paper, but that's a separate issue from abstract pattern recognition.