Solution to Problem 72


Congratulations to this week's winners

Nathan Pauli, Ray Kremer

Correct solutions were submitted by Cyril Terakopiantz, Philippe Fondanaiche, Aaron Kahn, Burkart Venzke and the proposer, Yan Fridman.


The only stable solutions are 3/4 and 0, these being the intersection of the line with equation y = x and the graph of the parabola described by the function 

f(x) = 4x (1 - x).

Every point which eventually stabilizes must map, after a certain finite number of iterations, to one of these.  So we need to look at inverses of these points. The inverse mapping is given by

Note that there will be two inverses at each stage (except for the point x = 1).

Starting from 3/4 we get  f -1(3/4) = 3/4, 1/4;  f -1(1/4) = ,
f -1() = , and then an easy induction shows successive inverses to be of the form

The same kind of calculation, starting from 0 gives f -1(0) = 1/2, f -1(1/2) =  and successive inverses now having the form

Aaron Kahn and Yan Fridman solved the problem through some clever changes of variable.

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Page last updated 16 November 1999.