Again received an overwhelming number of answers to this puzzle. People who answered it correctly included Pratik Poddar, Anant Aggarwal, Siddharth Mulherkar, Suman Saraf, P R Kumar and Amit Mittal. The puzzle had a simple trick to it, which is that the phrase we are trying to get to is embedded itself within the phrase.

I am reproducing the answer from Pratik Poddar.

Lets say that the value is x.

sqrt (12+x) = x

x^2 – x – 12 = 0

(x-4)(x+3) = 0

x = 4 (since x>0)

Hope you all enjoyed the puzzle!

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Btw, I had posted a comment on the original puzzle (not the solution) which appears to have been deleted. I am reproducing it again

If we assumed it was a converging series we could also have replaced the expressions like

x = sqrt(12+sqrt(12+x))

this would yield an equation of the 4th degree

x^4 – 24*x^2 – x + 132 = 0

which could be factorized as

(x-4)(x+3)(x^2 + x – 11) = 0

the two positive roots are 4 and (3*sqrt(5) – 1)/2

Similarly we could extend it to higher orders and find more positive roots. Wouldn’t all of them be satisfactory solutions?

Thoughts?

Suman, this one has stumped me, I did not think of a second order substitution. Even though the equation solves with these values, I am still having a hard time as the value is approx 2.8 which seems impossible.