We can actually solve Towers of Hanoi non-recursively by manipulating bits:

    max = 1 << no_of_discs;
    for (x = 1; x < max; x++)
        printf("move a disc from %d to %d\n", (x&x-1)%3, ((x|x-1)+1)%3);

where x&x-1 is smaller than x as much as (x|x-1)+1 is larger than x, by a 2-power corresponding to the least significant 1-bit in x.

Could you explain why that's true? - get that it works but don't understand why

I never constructed a proof either, but it would rest on the observation that the i'th smallest disc (i=1..no_of_discs) moves every 2^i-th step in the same direction (+1 mod 3 for odd i and -1 mod 3 for even i).

> It takes O(n*n) time to add the n integers.

Eh?

The time complexity assumption for addition is given earlier in the article.

From the article:

> We will be dealing with arbitrary precision integers (bignums) in the problem, so let us also make a few assumptions:

> Addition or subtraction of an m-bit integer and an n-bit integer (m <= n) takes O(n) time.

> Counting the number of 1-bits in an n-bit integer takes O(n) time.