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 Four Fours Fancy Formula! (Posted on 2004-12-24)
Find x if √(4+√(4-√(4+√(4-x))))=x.

 See The Solution Submitted by Old Original Oskar! Rating: 2.2857 (7 votes)

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 heuristic solution | Comment 1 of 6

By repeatedly replacing the value of x with the formula on the left side, x converges rapidly.  Starting with a guess of x=2, it converges thus:

`2.3007617081561682.3027616563776542.3027755406394622.3027756370577412.3027756377273122.3027756377319622.3027756377319942.3027756377319952.302775637731995...`

I squared the terminal number to see if this was the square root of an integer or of a rational number, which it was not, but it was exactly 3 higher than the number itself (it came out to 5.302775637731996, the difference in the final digit attributable to rounding differences).

So, solving x^2-3 = x:

x^2 - x - 3 = 0

x = (1 +/- sqrt(1 + 12)) / 2

As the number is positive, we get (1 + sqrt(13)) / 2.

The program:

DEFDBL A-Z
x = 2
FOR i = 1 TO 40
x = SQR(4 + SQR(4 - SQR(4 + SQR(4 - x))))
PRINT x
NEXT

 Posted by Charlie on 2004-12-24 15:10:29

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