Numerical solution:
1038733707Programs:
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) == 649987122332223332233 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:and provides exactly a recurrence relation and a dynamic programming approach.
www.reddit.com/r/dailyprogrammer/comments/8df7sm/20180419_challenge_357_intermediate_kolakoski/ might offer some reference implementations. It references a longer slide by Brent: maths-people.anu.edu.au/~brent/pd/Kolakoski-UNSW.pdf
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:
- pubs.sciepub.com/tjant/5/4/4/index.html Some Formulas for the Generalized Kolakoski Sequence Kol(a, b) by Abdallah Hammam. Maybe these identities could be useful.
ARC-AGI is a black hole for early retired tech and finance bros by 
Ciro Santilli 37 Created 2025-10-27 Updated 2025-12-13
Ciro Santilli 37 Created 2025-10-27 Updated 2025-12-13People who do cool open tech stuff when don't need money anymore are awesome:
- François Chollet, project founder: www.linkedin.com/in/fchollet/ 9 years at Google from 2015 to 2024. He founded ARC while he was still at Google though, so maybe doesn't count
- Cristiano Calgano from cristianoc/arc-agi-2-abstraction-dataset. Imperial College London researcher who founded a formal verification company and sold it to Facebook where he staid for 7 years
- Benjamin Crouzier from Tufa Labs
Benjamin has a masters in Computer Science and applied ML to quant finance previously, tufalabs.ai/team.html mentions:
Contains 120 DSL implementations for the
From another awesome retired tech bro that does this project for fun.
Cool website tracking the status of varios
Ciro Santilli's fork of ARC-DSL merging all pull requests needed to make tests run again on Ubuntu 25.04.
This interesting repo defines a set of input transformations that can be composed together into programs to generate the solve ARC problems.
It does not appear to have any program synthesis: it only defines the DSL and then provides manual solutions to the problems.
The README is lacking as usual, an overview of the files is:
- dsl.py: defines the transformations as Python functions
- solvers.py: defines solvers for the 400 ARC-AGI-1 training problems
Intended usage to run the solvers seems to be:Unfortunately this blows up on Ubuntu 25.04 on
git clone https://github.com/fchollet/ARC-AGI
cd ARC-AGI
git checkout 399030444e0ab0cc8b4e199870fb20b863846f34
git clone https://github.com/michaelhodel/arc-dsl
cd arc-dsl
git checkout 635de4902a5fb4e376f27333feaa396d3f5dfdcb
python main.pytest_mpapply apparently due to a Python 3.12 issue and the pull request github.com/michaelhodel/arc-dsl/pull/7 has been ignored for more than one year, so the project is largely dead.The rich guy behind Tufa Labs:
- github.com/pinouchon
- pinouchon.github.io
- www.youtube.com/benjamincrouzier He's also a wooden block fanatic. Funnily Ciro Santilli had seen his channel before knowing he was rich. www.youtube.com/watch?v=5HyYyXHSW74 shows probable home and could likely be geolocated.
- tufalabs.ai/team.html describes how he made his money:
Benjamin has a masters in Computer Science and applied ML to quant finance previously.
There are unlisted articles, also show them or only show them.