Redstone Clock — Solution¶

Comparing solutions¶

Tabulated and Euclidean GCD are peer methods. The former suits many queries when value-range tables are affordable; the reference uses Euclid for constant auxiliary space.

The problem setter’s solution also derives LCM from GCD, but first sieves the whole value range and splits each number into at most three query-friendly factors. Small factors are at most \(\sqrt U\) ; larger ones are primes handled by divisibility tests. GCDs between small values are tabulated. A query takes the GCD of each factor with the remaining part of the other number and divides out matched factors; their product is the full GCD without double-counting prime factors. The number of factors is bounded by a constant. Preprocessing takes \(O(U)\) time and space, followed by \(O(n)\) total query time in the fixed-word model.

The implementation used by Codex (GPT-6) generates queries in the same order and computes GCD directly with Euclid's algorithm. It uses \(O(n\log U)\) time and \(O(1)\) extra space. Removing value-range tables and preprocessing simplifies the implementation, but each query gains a logarithmic upper bound. With very many queries and sufficient table memory, the problem setter’s remains useful. This is an alternative solution.

Least common multiples¶

A positive time when both clocks finish a cycle must be a multiple of both periods. The earliest such time is

\[ \operatorname{lcm}(i,a)=\frac{i}{\gcd(i,a)}\,a \]

Compute the GCD for each query and XOR the LCM into the appropriate accumulator. Update \(a\) immediately after each query using its specified formula; the order of the two updates matters.

Correctness¶

The least common multiple is exactly the first positive time when both clocks finish a cycle. Iteration \(i\) processes query \(2i-1\) , updates the parameter, processes query \(2i\) , and updates again. This matches every step of the specified generation process, so the two accumulators contain exactly the odd-query and even-query answers.

Complexity and implementation¶

Let \(U\) bound the periods and initial parameter; here \(U=10^7\) . Each GCD takes \(O(\log U)\) time, giving total time \(O(n\log U)\) and auxiliary space \(O(1)\) under fixed-width integer arithmetic.

Both the LCM and \(a^2\) may exceed 32-bit range, so use 64-bit integers. Divide by the GCD before multiplying by \(a\) . Queries are generated online and need not be stored.