Block Tower: Generate Moves by Step Number¶
Codex (GPT-6) supplies and verifies the established step-number Hanoi construction in alternative.cpp. See the main editorial 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
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
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.