In case anyone wants to discuss the contest for Wolfram’s rule 30 on Community, please respond on this thread.
It’s about the center column of rule 30. There are three questions. The answer is a mathematical proof. There are cash prizes.
ArrayPlot[CellularAutomaton[30, {{1},0},100]]
I wanted to play with cellular automata for a while. When I saw the post about the contest, it finally got me into gear and I had some fun with rule 30:
From the abstract:
The usual pattern starting from a single cell with state 1 / black is examined. It is contructed from a richer structure which yields a progression of polynomials for diagonals from the right. It is shown that each diagonal from the left can be expressed by the progression of polynomials for diagonals from the right and how the connection between left and right diagonals forces the diagonals from the left to eventually become periodic.
The necessary and sufficient condition for period doubling of diagonals from the right is established. It is also shown that periods cannot decrease. The result suggests that predicting period doubling is computationally expensive.
I suppose there is nothing new in there?
Hi, I have an algebraic equation for RULE30, it runs on octave but would be great to see it iterated on Mathematica, #rule30 #equation #debian #gnu #linux #octave #nasm | Graham Medland
I also have a turmite equation (based on my langtons ant wave equation) which would be nice to see in Mathematica, could someone please help me ?
Hi Graham,
it seems that one needs a LinkedIn account to access that document. I suggest to change that.
Michael
Wolfram’s Elementary Cellular Automata
(Left) rule30 using my new equation ( can be done on a four function calculator )
(Middle) 4th order Z transform of occupancy matrix.
(Right) Power spectra of the 4th order co-effecients WOW! really exciting find 
It’s hard to tell if it’s new, Michael. Can you explain here a little more about what you’ve done?
Usually there will at least be a part of something like this that’s new.
Thanks!
Graham this looks neat. Can you explain a little more?
I understand that it’s not in Mathematica. If it’s one or two lines of math, we can translate it.
hi Todd,
what do you need to calculate a cell in a diagonal? The previous cell in that diagonal and two cells from the previous two diagonals. This enables one to define diagonals recursively. For diagonals from the right, the recursive definition leads to a simple picture: The previous two diagonals are combined with OR and the resulting values are combined with XOR - basically a summation over the combination up to the cell you need. It is this summation modulo 2 that causes the period doubling. I show that it will either double the basic period of the combination or leave it untouched. But combining the previous two diagonals with OR may decrease the basic period - so I also show that this cannot happen in diagonals from the right.
This is what may be new - I don’t know. The rest should be well known stuff - maybe looked at from a different viewpoint. Using a richter structure by replacing A OR B with A+B+A*B did not yield any really interesting results IMO but I left it in the article because it was my starting point.
Hi Todd, please email me, gmail.com@graham.medland
Graham
Thanks, yes, that sounds familiar from a talk Eric Rowland gave. It is pretty neat though, and yeah, a little hard to explain in words.
Hi everyone, can anyone tell me if the rule 30 prize is still running? I have been working very hard on it for about a year! I have left a comment on the offical page, which is still awaiting moderation, just getting a bit nervous that everyone has forgotton about it! Thank you, David
Hi everyone,
I submitted a solution paper to the Wolfram Rule 30 Prize around January 17, 2026 using the official “Submit a Solution” form, but I didn’t receive any confirmation email.
For those who have submitted before: do you typically get an automatic acknowledgment, or is it normal to hear nothing unless the committee needs clarification? Also, what kind of response timeline should I expect?
Thanks in advance!
Hi Tigran, I am still working on my rule 30 solution, I am not sure if the prize is still running as I dont see any activity there. want to meet up and chat about your ideas?
Hi David, thanks for the reply!
Good luck with your Rule 30 work, it’s definitely a deep and fascinating problem. I’m also not entirely sure about the current status of the prize, which is part of why I asked.
I’d be happy to chat and exchange ideas. Feel free to reach out and we can find a time that works.
Hi everyone,
I have been working heavily on the Rule 30 Prize problems and wanted to share a rigorous algebraic approach I just posted to the community.
I developed a framework that completely lifts Rule 30’s discrete Boolean logic into continuous integer calculus by using the binomial basis. By projecting this back to F_2 via Lucas’ Theorem, the automaton’s non-linear evolution translates into an exact subset OR-convolution over finite integer support sets, which we can call Sm
I have successfully compressed these support sets into masked dyadic blocks, which formally decouples spatial evaluation from temporal generation and allows the state of any cell to be queried in a strictly bounded O(logn) time if we have already computed Sm.
However, I have hit a wall and would love the input of the mathematicians here. My evaluation of Sm is still recursive. I am trying to determine if it is mathematically possible to find a closed-form (non-recursive) expression for Sm, or if the +1 carry-collisions introduced by the increment operator permanently shatter these subsets. If it is the latter, could this serve as the formal proof of Computational Irreducibility that the prize is looking for?
I have attached my full paper and Mathematica notebooks in my dedicated post. I would be honored if anyone here working on the contest would take a look and share their thoughts: https://community.wolfram.com/groups/-/m/t/3647733
Thanks
## Does an exact reconstruction of the Rule 30 cone contain a route to answering the challenges?
I have not used Wolfram products to explore the Rule 30 challenges. I don’t have a formal background for this work but I enjoy impossible problems. In the pursuit, I developed a formula capable of reproducing the possibly complete Rule 30 cone.
I think I may have come close to the solution, become confused or perhaps I have it and do not know it. In any case, I was happy to see the invitation to ask the human community what you think.
The formula works by flattening the cone row by row, reading each row from left to right.
Starting with (F(0)=1), for each index (n\geq1), let
𝑡=𝑙𝑓𝑙𝑜𝑜𝑟𝑠𝑞𝑟𝑡𝑛𝑟𝑓𝑙𝑜𝑜𝑟,𝑞𝑞𝑢𝑎𝑑𝑟=𝑛−𝑡2.
Here, (t) is the row number and (r) is the position within that row, both counted from zero. Perfect squares mark the beginnings of rows.
The three parent values are
Misplaced &
Misplaced &
and
Misplaced &
These conditions set any parent outside the preceding row to zero. The value at (n) is then
𝐹(𝑛)=𝐿𝑛𝑜𝑝𝑙𝑢𝑠(𝐶𝑛𝑙𝑜𝑟𝑅𝑛),
where (\oplus) means exclusive OR and (\lor) means OR.
The center column is selected at the pronic positions—the products of consecutive integers:
𝑐𝑡=𝐹𝑏𝑖𝑔𝑙(𝑡(𝑡+1)𝑏𝑖𝑔𝑟).
Im not familiar with the Wolfram software so I wrote a standalone Python program that compares this formula with conventional Rule 30 evolution. It checks the 65,536 cells of the first 256 complete rows and also checks the first 4,096 center states separately, all show exact agreement.
The program uses only the Python standard library, I posted it to github here:
These are finite checks, so I only assume that the agreement continues. I was satisfied that I had broken the problem out of its single origin enclosure so I could approach the problem from an alternate origin. Which may have been the challenge itself. Then I became confused and went to bed.
My question is:
Can this formula help prove that the center column never becomes eventually periodic, has equal limiting frequencies of zeros and ones, and requires at least linear time to compute its (n)th bit? Or have I simply expressed Rule 30 differently?
