# Block Tower: Generate Moves by Step Number

Codex (GPT-6) supplies and verifies the established step-number Hanoi construction in [alternative.cpp](alternative.cpp). See the [main editorial](solution.en.md) for the preferred recursion and optimality proof.

## Which block moves now

Here the top, smallest block is $d=1$ and the bottom block is $d=n$ . Number moves from $t=1$ and set

$$d=1+\nu_2(t)$$

where $\nu_2(t)$ counts factors of $2$ in $t$ . This is the block moved now, with input-name index $n-d$ .

Number platforms $A,B,C$ as $0,1,2$ . Each block moves in one fixed cyclic direction. If $n-d$ is even, add $1$ modulo $3$ per move; otherwise add $2$ modulo $3$ . Call this increment $\delta_d$ .

Block $d$ moves at steps $t=(2k+1)2^{d-1}$ , after $k=\lfloor t/2^d\rfloor$ earlier moves. Its source and destination are therefore

$$a=k\delta_d\bmod3,\qquad b=(a+\delta_d)\bmod3$$

Enumerate $t=1,\ldots,2^n-1$ and print the corresponding instruction.

## Agreement with the recursive construction

Induct on $n$ . The middle step $t=2^{n-1}$ moves only the bottom block, from $A\to B$ . The two surrounding halves transfer the upper tower from $A$ to $C$ and from $C$ to $B$ .

Each subproblem exchanges platform roles relative to the standard $n-1$ problem. This reverses the smaller blocks' cyclic directions, exactly matching the change in parity of $n-d$ . The second half continues the earlier move counts. For each $d<n$ , occurrence positions are still odd multiples of $2^{d-1}$ , while the largest block occupies the middle. Thus block identities, accumulated move counts and platform directions agree with the recursive sequence. The output is legal and optimal.

## Complexity and use

The implementation obtains $\nu_2(t)$ by repeated division by $2$ . Across all steps, these extra divisions total less than $2^n$ : there are $\lfloor(2^n-1)/2^j\rfloor$ steps with at least $j$ factors, and the sum over $j\ge1$ is less than $2^n$ .

Time remains $O(H+Z+n)$ , where $H$ counts input name characters and $Z$ counts output characters. Names use $O(H+n)$ space; all other state is constant. This avoids a recursion stack without changing the overall space bound including names. It is a backup for direct indexed generation; see the [main timing results](solution.en.md#measured-results).