IBM Research | Ponder This | January 2004 solutions

# Ponder This

## January 2004

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Solution:

This technique is known as Pollard's rho method; it was developed by John Pollard as a tool for integer factorization.

Let f(x) be the location of the array at address x.

a := f(f(n))
b := f(n)
do while not (a=b)
a:=f(f(a))
b:=f(b)
end
/* Now a=b can be reached at either 2k or k steps from n, */
/* where k is some integer between 1 and n. */
a:=n
do while not (a=b)
a:=f(a)
b:=f(b)
end
print a

The pattern that you get, starting from n and repeatedly applying f (doing the table lookup), is a string of some length L>0 leading to a cycle of some length M, with L+M<n. (L is nonzero because n is outside the range of f.)
It is shaped like the Greek letter rho, hence its name. Our stopping count k is the least multiple of M greater than or equal to L, so that L \leq k < L+M < n.

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