Project Euler problem 943 Created 2025-10-27 Updated 2025-11-23
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Numerical solution:
1038733707
Programs:
A naive T in Python is:
from collections import deque

def T(a: int, b: int, N: int) -> int:
    total = a
    q = deque([a] * (a - 1))
    is_a = False
    for i in range(N - 1):
        cur = q.popleft()
        total += cur
        q.extend([a if is_a else b] * cur)
        is_a = not is_a
    return total

assert T(2, 3, 10) == 25
assert T(4, 2, 10**4) == 30004
assert T(5, 8, 10**6) == 6499871
which passes the tests, but takes half a second on PyPy. So clearly it is not going to work for 22332223332233 which has 14 digits.
Maybe if T is optimized enough, then we can just bruteforce over the ~40k possible sum ranges 2 to 223. 1 second would mean 14 hours to do them all, so bruteforce but doable. Otherwise another optimization step may be needed at that level as well: one wonders if multiple sums can be factored out, or if the modularity can of the answer can help in a meaningful way. The first solver was ecnerwala using C/C++ in 1 hour, so there must be another insight missing, unless they have access to a supercomputer.
The first idea that comes to mind to try and optimize T is that this is a dynamic programming, but then the question is what is the recurrence relation.
The sequence appears to be a generalization of the Kolakoski sequence but to numbers other than 1 and 2, also known as the Generalized Kolakoski sequence.
maths-people.anu.edu.au/~brent/pd/Kolakoski-ACCMCC.pdf "A fast algorithm for the Kolakoski sequence" might provide the solution, the paper says:
It is conjectured that the algorithm runs in time and space , where
and provides exactly a recurrence relation and a dynamic programming approach.
www.reddit.com/r/algorithms/comments/8cv3se/kolakoski_sequence/ asks for an implementation but no one gave anything. Dupe question: math.stackexchange.com/questions/2740997/kolakoski-sequence contains an answer with Python and Rust code but just for the original 1,2 case.
github.com/runbobby/Kolakoski has some C++ code but it is not well documented so it's not immediately easy to understand what it actually does. It does appear to consider the m n case however.
Bibligraphy:
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Project Euler problem style 2026-01-30
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Project Euler problems typically involve finding or proving and then using a lemma that makes computation of the solution feasible without brute force. There is often an obvious brute force approach, but the pick problem sizes large enough such that it is just not fast enough, but the non-brute-force is.
As such, they live in the intersection of mathematics and computer science.
news.ycombinator.com/item?id=7057408 which is mega high on Google says:
I love project euler, but I've come to the realization that its purpose is to beat programmers soundly about the head and neck with a big math stick. At work last week, we were working on project euler at lunch, and had the one CS PhD in our midst not jumped up and explained the chinese remainder theorem to us, we wouldn't have had a chance.
In many cases, the efficient solution involves dynamic programming.
There are also a set of problems which are very numerical analysis in nature and require the approximation of some real number to a given precision. These are often very fiddly as I doubt most people can prove that their chosen hyperparameters guarantee the required precision.
Many problems ask for solution modulo some number. In general, this is only so that C/C++ users won't have to resort to using an arbitrary-precision arithmetic library and be able to fit everything into uint64 instead. Maybe it also helps the judge system slightly having smaller strings to compare. The final modulos usually don't add any insight to the problems.
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