Mastodawn

In-Sight Publishing Jan 23, 2025

Breaking Bread with Marco Ripà on Iterated Exponentiation and the Nine Dots Problem (4)

Scott Douglas Jacobsen
In-Sight Publishing, Fort Langley, British Columbia, Canada

Correspondence: Scott Douglas Jacobsen (Email: [email protected])

Received: January 16, 2025
Accepted: N/A
Published: January 22, 2025

Abstract

This interview provides an in-depth exploration of Marco Ripà’s innovative contributions to discrete mathematics as an independent researcher and autodidact. Conducted by Scott Douglas Jacobsen, the conversation delves into Ripà’s dual focus on the modular properties of integer tetration and the development of minimal covering paths in high-dimensional grids, including his work on the Nine Dots Puzzle and generalized knight’s tours. Ripà discusses his fascination with iterated exponentiation, the concept of congruence speed, and his creation of over 100 sequences in the OEIS. The dialogue also covers Ripà’s formula for generating infinitely many c-th perfect powers whose constant congruence speed is also equal to c, for any positive integer c, his counterexamples to established theorems, and the potential applications of his research in cryptography and Ramsey theory. Additionally, Ripà outlines his future research directions, including collaborations on Euclidean knight’s tours and optimization problems in high-dimensional grids. The interview highlights Ripà’s unique approach to mathematics, characterized by innovative algorithms, constructive proofs, and a commitment to expanding the boundaries of combinatorics, number theory, and graph theory.

Keywords: Cryptography, Combinatorics, Congruence Speed, Discrete Mathematics, Graph Theory, High-Dimensional Grids, Hypercubes, Integer Tetration, Knight’s Tour, Minimal Covering Paths, Nine Dots Puzzle, Number Theory, OEIS, Perfect Powers, Ramsey Theory

Introduction

In this comprehensive interview, conducted on January 16, 2025, Scott Douglas Jacobsen engages with Marco Ripà, a self-taught mathematician and independent researcher renowned for his extensive work in discrete mathematics. Ripà has dedicated the past decade to exploring the modular properties of integer tetration and developing minimal covering paths in high-dimensional point grids, including innovative solutions to the classic Nine Dots Puzzle and its higher-dimensional generalizations. With over 100 sequences contributed to the Online Encyclopedia of Integer Sequences (OEIS) since 2011, Ripà has established himself as a prolific figure in combinatorics and number theory. The conversation examines Ripà’s early fascination with iterated exponentiation, his groundbreaking concepts such as congruence speed, and his contributions to the understanding of knight’s tours in multidimensional chessboards. Additionally, Ripà shares his insights into generating perfect powers with constant congruence speeds, his challenges to existing mathematical theorems, and the broader implications of his research. The interview also touches on Ripà’s future projects, including collaborations on Euclidean knight’s tours and optimization problems in high-dimensional grids, positioning him as a leading innovator in the evolving landscape of mathematical research.

Main Text (Interview)

Interviewer: Scott Douglas Jacobsen

Interviewee: Marco Ripà

Section 1: Introduction to Marco Ripà’s Research

Scott Douglas Jacobsen: Over the past decade, you’ve pursued a unique dual-aspect focus. One on the modular properties of integer tetration or iterated exponentiation: how rightmost digits stabilize with growth in power tower and introduction of congruence speed. Another on minimal covering paths in high-dimensional point grids: emphasis on the 3×3 Nine Dots Puzzle with a clockwise-algorithm to minimize straight segments needed to cover all points. Other work has covered perfect powers, number-theoretic conjectures, and Hamiltonian paths in higher-dimensional hypercubes. So, what is your background interest as an independent researcher and autodidact in mathematics?

Section 2: Fascination with Iterated Exponentiation

Marco Ripà: That’s fundamentally correct. Given that, in the same period, I’ve contributed over 100 sequences published on the OEIS (180+ since 2011), e.g.:

Ripà’s sequences for the OEIS

Single author of (175): A176942, A180346, A181073, A181129, A187602, A187603, A187604, A187605, A187613, A187628, A187636, A206636, A220090, A225227, A244056, A260586, A260751, A260819, A260820, A260821, A260823, A260825, A260939, A260953, A260955, A260982, A261039, A261040, A261066, A261067, A261071, A261116, A261140, A261149, A261150, A261151, A261152, A261193, A261547, A306658, A306700, A306744, A307379, A317163, A317164, A317255, A317259, A317824, A317903, A317905, A317914, A318165, A318478, A319259, A320523, A321130, A321131, A335112, A335113, A335114, A337392, A337833, A337836, A339313, A340009, A340012, A340036, A340315, A340319, A340345, A340841, A349425, A351663, A351727, A352396, A352991, A353025, A353152, A353238, A353246, A354209, A354619, A354959, A355417, A355424, A355461, A356022, A356023, A356562, A356751, A356756, A356946, A356987, A357055, A357056, A357969, A358030, A358361, A359201, A359187, A359202, A359662, A359740, A359855, A360178, A360180, A360270, A360750, A361006, A361010, A361011, A361100, A361918, A361352, A362000, A362004, A362529, A362530, A362590, A363746, A363755, A363766, A363980, A364270, A364271, A364711, A364789, A364837, A364855, A365689, A365707, A365935, A365936, A365937, A368008, A368009, A368146, A368476, A369624, A369771, A369826, A370211, A370255, A370532, A370775, A371048, A371074, A371078, A371129, A371671, A371720, A372490, A373205, A373206, A373386, A373387, A373537, A374148, A374149, A374224, A374260, A374883, A374948, A374949, A376446, A376838, A376842, A376883, A377124, A377126, A378419, A378421, A379243, A379906, A380031.

First author of (7): A306780, A307383, A307384, A307385, A355420, A356805, A364806.

Co-author of (3): A181373, A352329, A356810.

More terms added to the original sequence (1): A254276.

Reference: The On-Line Encyclopedia of Integer Sequences (OEIS), https://oeis.org/search?q=author%3Arip%C3%A0&sort.

We could synthetically say that my focus as an Independent Researcher (100% self-taught in mathematics) is discrete mathematics. For example, let us consider my preprint Shortest polygonal chains covering each planar square grid: the original proof of Lemma 2.1 in the Appendix transposes a graph theory problem into a Diophantine equation, which is discrete number theory in the end.

Research-level mathematics is incredibly challenging for an autodidact to navigate across multiple fields. I chose combinatorics and number theory because many open problems lie within these domains (as you can see by visiting Open Problem Garden) and are not as technically demanding as branches that merge algebra and topology. Additionally, I have been fascinated by integers since I was very young. So here I am.

Section 3: Personal Motivation Behind the Nine Dots Puzzle

Jacobsen: How did you become fascinated by iterated exponentiation?

Ripà: I grew up as a lonely child. My mom worked in her toy store, and I spent countless hours there, often bored with nothing to do. In the late ’80s, my best friend in that store was an old calculator with only the four basic arithmetic operations. I discovered that if I typed a number, pressed “×” twice, and then repeatedly hit “=”, the result grew rapidly, filling the entire screen. I was fascinated by the patterns in the final digits.
About 25 years later, I discovered WolframAlpha and revisited this “game,” replacing “×” with “^”, thus exploring power towers instead of iterated multiplications. I suddenly noticed how the final digits stabilized, and then I realized that even the digit next to the leftmost stable digit exhibited non-random behavior, eventually entering a periodic loop. It reminded me of those days in my mom’s store.

Delving deeper into these patterns, I noticed that the number of new frozen digits after each iteration became constant at some point. This observation inspired me to study the topic more formally. I wrote a book in 2011 and eventually published the first peer-reviewed paper on the subject, titled “On the constant congruence speed of tetration”, in 2020.

Section 4: Exploring Integer Tetration and Modularity

Jacobsen: What is iterated exponentiation or modularity of integer tetration?

Ripà: I focused my work on integer power towers in radix-10, the familiar decimal numeral system (bring closure to the countless hours spent by that lonely child repeatedly pressing “×” followed by “=” over and over), where all the entries are the same given positive integer.

These are called “tetrations” (the next operation above exponentiation, also called hyper-4), where we raise a given number “a” to itself “b” times. When we write a^^b, we mean a^a^a^…^a, starting from the top and working downwards. For instance, 3^^3=3^3^3=3^(3^3)=3^(27)=7625597484987, instead of (3^3)^3=27^3=19683 (which is not particularly interesting from a mathematical point of view). The latter is sometimes referred to as “weak tetration”, but mathematicians like to make simple things complex, so why be surprised by that choice? (Just joking — we are serious thinkers, n’est-ce pas?).

Section 5: Minimal Covering Paths in High-Dimensional Grids

Jacobsen: What are minimal covering paths in high-dimensional point grids?

Ripà: Here we are in the field of combinatorics and graph theory. We are considering grids of points in a Euclidean k-dimensional space, so we can properly define those sets of n_1⋅n_2⋅⋅⋅n_k points through the Cartesian product {0,1,2,…,(n_1-1)}×{0,1,2,…,(n_2-1)}×⋯×{0,1,2,…,(n_k-1)}.

A minimal covering path for these grids is a simple polygonal chain with the minimum number of edges that fits all the given n_1⋅n_2⋅⋅⋅n_k points. If we visit any of those points more than once, we have a trail and not a path (paths are also trails, but the reverse is not generally true). We can solve the Nine Dots Problem by drawing a trail, but we can also find solutions that are paths…

Section 6: The Classical Nine Dots Problem Explained

Jacobsen: What is the classical Nine Dots Problem?

Ripà: In my previous answer, I assumed that everybody here already knows the rules of the most famous divergent-thinking puzzle ever — the one that inspired the “thinking outside the box” idiom (it appeared in Sam Loyd’s Encyclopedia of Puzzles, page 301, more than one century ago).

Basically, we can describe it mathematically as follows:

take the 3×3 grid {0,1,2}×{0,1,2} and then consider the square [0,3]×[0,3]. Now, try to join all the points by drawing a polygonal chain with no more than four edges, entirely contained inside the mentioned square [0,3]×[0,3].

Section 7: Generalizing the Nine Dots Problem to Higher Dimensions

Jacobsen: What drew you to it?

Ripà: My former girlfriend, at the time, challenged me to solve it. I found a solution in about 30 seconds; she thought I already knew the correct answer. She lived far from my house, so I spent the whole journey thinking about harder versions of the basic puzzle while the train was traveling back to Rome (many great ideas came during train journeys, from Mickey Mouse to Harry Potter… unfortunately, I wasn’t as lucky as Walt Disney or J. K. Rowling; and that’s all, folks).

Jacobsen: Why generalize it to higher dimensions?

Ripà: Because it is the most natural way to generalize the classic puzzle, as the planar extension to n_1×n_2 grids admits a trivial solution and was proved by others (even though I only discovered this a few years later).

Section 8: Key Insights into k-Dimensional Solutions

Jacobsen: How would you summarize the key insight behind your general k-dimensional solution to the Nine Dots Problem?

Ripà: Although I reached the solution independently by gradually reducing the gap between the proved upper and lower bounds, a fascinating pattern was revealed at the very end: the clockwise-algorithm simply returns the k-dimensional generalization of the solution to the classic Nine Dots Puzzle by Loyd. To understand the concept, let us take the k=3 case: the solution is a trail consisting of 13 line segments in space, but when you view it from the right perspective, you only see the Nine Dots Puzzle solution drawn on a piece of paper. Here’s a 4D video showing a few solutions returned by my clockwise-algorithm (up to 4 dimensions).

Section 9: Constructing k-Dimensional Covering Paths

Jacobsen: What are the main mathematical steps or ideas in constructing that k-dimensional solution?

Ripà: The proof starts with establishing the trivial lower bound (n^k-1)/(n-1) for any n×n×⋯×n grid, where n is an integer greater than 2. If n=3 is given, then the lower bound becomes (3^k-1)/(3-1). Our goal is to find, for any given integer k>0, a constructive algorithm that produces trails with (3^k-1)/2 lines joining all the 3^k points of the set {0,1,2}^k in ℝ^k.

Section 10: Elegance in the Generalized Nine Dots Solution

Jacobsen: What is the most elegant part of the Nine Dots generalization?

Ripà: In my opinion, it is the way the solution for the k-1 case is used to produce a valid minimum-link trail for the k-dimensional case. When 3^(k -1)-1 points are left, the trail with 3^(k-1)/2 line segments that covers {0,1,2}^(k -1) is applied to the remaining unvisited points. In general, each solution produced by the clockwise-algorithm can be viewed from the right perspective to reveal a trail of 3^(k-1)/2 segments covering a {0,1,2}^(k-1) grid. For example, in a 3D solution of this kind, you will see Loyd’s expected solution for the classic Nine Dots Puzzle, and again, if we look at it the same way, we will see just a line segment fitting 3 collinear points… which perfectly overlaps with the solution for the case k=1.

Moreover, as k grows, the number of different optimal trails increases. Each k-dimensional solution obtained by applying the clockwise-algorithm to an optimal trail for the (k-1)-dimensional configuration retains the same features of the input trail with 3^(k-1)/2 segments. This process extends in the same way as the four-line solution for the Nine Dots Problem generalizes to a line segment fitting the points {(0), (1), (2)}: it performs a 3-step circuit returning to the starting point, leaving exactly 3^1-1 collinear points, allowing a line segment to connect to the farthest grid point. Essentially, the solution replicates the line segment fitting {(0), (1), (2)}, and while we’re ignoring stretching, lengths are irrelevant for such mathematical problems where graph isomorphisms are the focus.

Section 11: Knight’s Tours on Higher-Dimensional Chessboards

Jacobsen: On the knight’s tours, what is the significance of proving there are infinitely many (Euclidean) tours on 2×2×⋯×2 chessboards?

Ripà: In the strictest sense, a knight’s tour on a given n×n×⋯×n, k-dimensional, chessboard requires moving a knight (following the chess knight move rule, whatever it entails) from a chosen cell back to the starting point (performing a closed knight’s tour), visiting all the other cells with the first n^(k-1) moves, without landing on the same cell more than once. Proving the existence of infinitely many Euclidean tours on such chessboards shows that this problem admits solutions under the natural generalization of the knight’s move to higher dimensions.

Section 12: Defining Knight’s Moves in Higher Dimensions

Jacobsen: What does a “knight’s move” mean in higher dimensions?

Ripà: Good question, thank you for asking. In my opinion, chess rules are determined by FIDE (The International Chess Federation), which has defined chess regulations for over 100 years, including the knight’s move. Article 3.6 of the FIDE Handbook, Section E/01 – Laws of Chess (see https://handbook.fide.com/chapter/E012023), states: “The knight may move to one of the squares nearest to that on which it stands but not on the same rank, file or diagonal”.

This definition implies that the knight moves to a cell at an exact Euclidean distance of sqrt(5) units from the starting point. Here, a “unit” is the distance between the centers of two adjacent cells (e.g., A1 and A2, or C6 and D6, on the classic 8×8 chessboard).

This conclusion follows directly from pure logic: if we were to restrict knights in higher dimensions to only perform “L-shaped” moves, we would wrongly admit moves that violate the geometric constraint. For instance, consider a knight placed at the center of the 3×3×3 chessboard {0,1,2}×{0,1,2}×{0,1,2}. Based on the “rank, file, or diagonal” rule alone, one might claim that the knight could move from (1,1,1) to (1,2,2) since (1,2,2) does not lie on the same rank, file, or diagonal as (1,1,1), but it is trivial to point out that the Euclidean distance between (1,1,1) and (1,2,2) is sqrt(2), which is smaller than sqrt(5). By correctly interpreting the knight’s move as requiring an exact Euclidean distance of sqrt(5), we ensure that the concept logically extends to higher dimensions without inconsistencies.

Section 13: Counterexamples to Existing Theorems on Knight’s Tours

Jacobsen: How does your result provide a counterexample to Theorem 3 in the Erde et al. (2012) paper?

Ripà: If the knight’s move as defined by FIDE is correct, then the rule can be stated as: “The knight may move to any cell at a Euclidean distance of exactly sqrt(5) units from the starting cell”. Now, I have constructively proved the existence of closed knight’s tours on any 2×2×⋯×2 chessboard with at least 64 cells. More specifically, such tours exist on all 2×2×⋯×2 chessboard with 2^k cells for k>5 (k≥6 is a sufficient and necessary condition).
Theorem 3 of the mentioned 2012 paper states that no closed knight’s tours exist on any n×n×⋯×n chessboard where n<3. However, my results provide constructive counterexamples under the assumption that FIDE’s definition of the knight’s move is authoritative.

The paper by Erde et al. does not explicitly define how a knight’s move generalizes to higher dimensions. It is likely the authors assumed L-shaped moves only, but this assumption no longer holds as k goes above 4. If someone disagrees with my construction, they must explain why a move from (1,1,1) to (0,0,1) would not be allowed according to FIDE’s definition of the knight’s move (by Article 3.6 of the Handbook, Section E/01).

Section 14: Distinct Approaches to Nine Dots and Knight’s Tour Problems

Jacobsen: Are you using a unifying technique or approach for the Nine Dots and Knight’s Tour problems?

Ripà: No, I am not. The approaches are entirely independent, relying on direct proofs. My constructive solution for the 3×3×⋯×3 Dots Problem (the clockwise-algorithm) does not describe a Hamiltonian cycle or even a path. Instead, it constructs minimum-link trails with (3^k-1)/2 edges joining all 3^k points.

Section 15: Generating Infinitely Many c-th Perfect Powers

Jacobsen: What is your formula for generating infinitely many c-th perfect powers?

Ripà: My formula not only generates infinitely many perfect powers of degree exactly c, but also ensures that these perfect powers are characterized by a constant congruence speed equal to c. This result is a particular case of what I consider the most elegant equation I have ever proved in number theory.

In detail, let V(…) denote the constant congruence speed of the argument, and let v_5(…) and v_2(…) indicate the number of times that 5 and 2 divide the argument. Let c, k, and t be positive integers such that t>min{v_2(c), v_5(c)}+1. Then, the formula V((10^(k+t)+10^(t-min{v_2(c), v_5(c)})+1)^c)=t holds for every pair (c, t), as k=1, 2, 3, … does not affect the result. The expression 10^(k+t)+10^(t-min{v_2(c), v_5(c)})+1 is constructed to be a multiple of 3 but not divisible by 3^2, ensuring that (10^(k+t)+10^(t-min{v_2(c), v_5(c)})+1)^c is always a perfect power of degree exactly c. The congruence speed formula guarantees that V((10^(k+t)+10^(t-min{v_2(c), v_5(c)})+1)^c)=t, which remains valid when we assign t:=c.
Finally, this leads to V((10^(k+c)+10^(c-min{v_2(c), v_5(c)})+1)^c)=c, providing infinitely many c-th perfect powers with the desired properties.

Section 16: Ensuring Constant Congruence Speed in Perfect Powers

Jacobsen: How does your formula ensure a constant congruence speed rather than a variable congruence speed for these c-th powers?

Ripà: The proof involves some technical special cases that we can safely omit here for simplicity. Let t>min{v_2(c), v_5(c)}+1 and assume c>1. The constant congruence speed of any positive integer greater than 1 and not a multiple of 10 is itself a positive integer. In collaboration with Luca Onnis, I derived the general formula to compute the constant congruence speed of any such tetration base. This formula is provided as Equation (3) in Number of stable digits of any integer tetration.

We observe that 10^(k+t)+10^(t-min{v_2(c), v_5(c)})+1 is a multiple of 3 (as the sum of its digits is 3) but not divisible by 3^2. Additionally, it ends 01. Therefore, we only need to consider the first line of Equation (3).
Then, using the Lifting The Exponent (LTE) lemma, I proved Lemma 2 in On the relation between perfect powers and tetration frozen digits, from which Theorem 2 follows.
These results are sufficient to prove the general equation: V((10^(k+t)+10^t+1)^c)=t+min{v_2(c), v_5(c)}.

We can safely replace t with t-min{v_2(c), v_5(c)} (as t-min{v_2(c), v_5(c)}≥2 by hypothesis).

This gives:

V((10^(k+t-min{v_2(c), v_5(c)})+10^(t-min{v_2(c), v_5(c)})+1)^c)=t. Since the value of the positive integer k does not affect the result, we simplify this to: V((10^(k+t)+10^(t-min{v_2(c), v_5(c)})+1)^c)=t.

Finally, when t:=c is given, we have V((10^(k+c)+10^(c-min{v_2(c), v_5(c)})+1)^c)=c, as c≥min{v_2(c), v_5(5)}+2 holds for any c>1) (Q.E.D.).

Section 17: Potential Applications Beyond Pure Mathematics

Jacobsen: Do you see any potential applications or implications of these formulas and proofs outside of pure mathematics?

Ripà: It’s hard to say definitively, but I’ve outlined a few ideas in the introduction of “On the relation between perfect powers and tetration frozen digits”. One potential application lies in cryptography. For example, I could challenge you to the following game:

Me: Choose a positive integer greater than 1 and not ending in 0, as large as you wish.

You: I choose “x”.

Me: Here’s a tetration base whose constant congruence speed is not (yet) stable at height x and whose congruence speed never matches its constant value at heights 1 to x.

For instance, if you choose x=103, I could construct a base such as:

[Ed. Numeric sequence immediately below should be taken as one line or complete unseparated sequence.]

4355257089251996605256803858446960842587857122227…

…47129609220545115161423787862411847003581666295807

which can then generate infinitely many tetration bases with similar properties.

Moreover, the congruence speed formula can solve a particular class of problems related to Ramsey theory. For instance, I used it to answer an open question asking to determine how many iterations of Hensel’s (lifting) lemma you can do in order to keep getting one more (new) ending digit of Graham’s number at a time… it is sufficient to invoke the first Lemma of my old paper “On the constant congruence speed of tetration”, but I proved the result by using the general formula. The result shows that Graham’s number, G, has exactly slog_3(G)-1 stable digits, where slog is the super-logarithm.

Furthermore, in my preprint “Graham’s number stable digits: an exact solution”, I showed that the difference between the (slog_3(G))-th least significant digit of G and that of 3^G is exactly 4 (where nobody is actually able to predict the first digit of G in the decimal numeral system — see the MathOverflow discussion “The problem of finding the first digit in Graham’s number”).

Section 18: Approaches to Identifying Counterexamples

Jacobsen: How do you typically approach “established” theorems or conjectures with an eye to finding counterexamples?

Ripà: I usually attempt to find counterexamples to any mathematical result I read, as a way to test my own understanding and reading comprehension. However, this was not the case with the mentioned Erde’s paper. At that time, I was working on a lengthy preprint about metric spaces in generalized chess (Metric spaces in chess and international chess pieces graph diameters), which required a deeper analysis of how chess pieces move can be defined.

Actually, the strictest logical solution I derived was the Euclidean knight definition and, you know, chess knights do not understand advanced mathematics, nor can they read published theorems… they simply move around any k-dimensional 2×2×⋯×2 grid as long as k>5, eventually returning to their starting cell aftere their (2^k)-th move, each jump spanning sqrt(5) chessboard units. Whether or not this constitutes counterexample to the mentioned theorem depends on the assumptions we make at the outset. In my humble opinion, if a mathematical object is not rigorously defined, theorems built upon it become vulnerable to such ambiguities. To clarify further, I believe even Euler’s concept of a knight aligned with the common understanding: one restricted to performing L-shaped moves in any number of dimensions.

Section 19: Future Directions and Research Expansions

Jacobsen: What directions or expansions do you envision for each of these results?

Ripà: Currently, Gabriele Di Pietro (a mathematics teacher) and I are further generalizing my initial breakthrough on Euclidean knight’s tours, extending the results to other fairy chess leapers (chess pieces with non-standard movement patterns that moves directly to a square a fixed distance away). These developments are detailed in the preprint On the existence of Hamiltonian cycles in hypercubes.

Regarding optimization problems on regular grids, another independent Italian researcher is working to improve the bounds I provided for the general n_1×n_2×⋯×n_k points configurations. Meanwhile, I have summarized the best current results for n×n×⋯×n grids in the preprint General conjecture on the optimal covering trails for any 𝑘 -dimensional cubic lattice (v3). In Optimal cycles enclosing all the nodes of a k-dimensional hypercube (2022), Roberto Rinaldi and I have also proved the exact solution for the (k-dimensional) 2×2×⋯×2 case, showing that 3⋅2^(k-2) links are sufficient to cover the grid with a cycle (not just a trail, circuit, or path). However, the 4×4×⋯×4 problem is still open, for any k≥3 (I’ve only been able to prove that the 4×4×4 grid can be covered with a path consisting of 23 links, while no trail with less than 21 links can do the same).

Lastly, Gabriele and I plan to develop a more compact notation to express the congruence speed (not only its constant value) and the phase shift (not only its asymptotic value) of any integer, including multiples of 10. Although I have already published preprints with all the necessary formulae, this new notation will improve readability and facilitate future research on the topic.

To encourage recreational exploration of this area, I have provided Twelve Python Programs to Help Readers Test Peculiar Properties of Integer Tetration that allow anyone to experiment with congruence speed and phase shift. As part of this effort, I also issued a challenge to my YouTube followers to solve at least three out of five problems within 2.5 hours. None of the participants managed to pass the test on January 3, 2025. For reference, the full text of the challenge and solutions can be found here: Five Hard Problems with a Simple Solution).

Section 20: Influence on Combinatorics, Number Theory, and Geometry

Jacobsen: How do you see your work influencing other areas of combinatorics, number theory, or geometry?

Ripà: As I mentioned earlier, I believe that the discovery of the constancy of the congruence speed of tetration (and the related congruence speed formula) holds great untapped potential. It could also find applications in areas such as Ramsey theory. Exploring congruence speed-related problems is a fascinating endeavor, particularly for individuals with strong analytical skills and a high level of pattern recognition. I coined the term “speed” because the height of the tetration can be interpreted as time, and the total number of stable digits as the “distance” traveled during the first “b” iterations. Consequently, the congruence speed maps the rate of frozen digits’ growth: from height 1 to 2, from height 2 to 3, and so forth. At certain points, depending on the base of tetration under consideration, this value stabilizes at the constant congruence speed value, as long as the given tetration base doesn’t end in 0.

Interestingly, the hyperoperator pentation — which is to tetration what tetration is to exponentiation — might similarly exhibit a constant “congruence acceleration” (assuming an integer pentation base greater than 1 and not divisible by 10).
Keeping the focus on the skills that led me to study tetration’s last digits, the classic Nine Dots Puzzle’s extension into three dimensions could serve as a useful psychometric tool for testing advanced pattern recognition. I’ve already created a similar challenge focused on two-dimensional grids, which I believe is well-suited for individuals with IQs in the 140+ (S.D. 15) range. You can find it here: DOTS Rev – A Real IQ Challenge for the High Range. Enjoy!

Section 21: Guidance for Researchers Building on Ripà’s Findings

Jacobsen: Finally, if someone wanted to explore or build upon your findings, where would you recommend they begin?

Ripà: It’s a personal choice, but my advice is to start with our more accessible papers on each topic. Here’s a roadmap:

Congruence speed and tetration:

Although the core paper is “The congruence speed formula”, since you need it to prove the results stated in “Number of stable digits of any integer tetration”, I suggest to begin from the latter. You may skip the complex proof involving the homomorphism with the commutative ring of 10-adic integers, as it delves into advanced topics like the 15 10-adic solutions of the fundamental fifth-degree equation y^5=y.

This introduces the congruence speed formula as Equations (3) and (16) and provides results that are easier to grasp. Then, read “Graham’s number stable digits: an exact solution”, which is far easier than “On the relation between perfect powers and tetration frozen digits”. Make sure to understand “Number of stable digits of any integer tetration” first, as it lays the groundwork.

For a deeper dive, explore Sections 3 and 4 of “The congruence speed formula”. Section 4, which proves the existence of infinitely many prime numbers characterized by any positive constant congruence speed by invoking the powerful Dirichlet’s theorem on primes in arithmetic progressions, is particularly fascinating. Finally, for the congruence speed formula of all the bases that are multiples of 10, check out the short and accessible Congruence speed of tetration bases ending with 0.

Optimization problems on regular grids:

Start with “Minimum-Link Covering Trails for any Hypercubic Lattice” (version 5) for lower bounds and “General conjecture on the optimal covering trails for any 𝑘-dimensional cubic lattice” for upper bounds.

For open research lines, “General uncrossing covering paths inside the axis-aligned bounding box” is an excellent starting point. It introduces new optimization problems and opens avenues to create many OEIS sequences. This paper is the culmination of a trilogy that began with “Solving the 106 years old 3^k points problem with the clockwise-algorithm”.

Euclidean knight’s tours and generalized mulitidimensional chess pieces:

Begin with the concise and easy-to-follow “Proving the existence of Euclidean knight’s tours on n×n×⋯×n chessboards for n<4”.

Next, explore “Metric spaces in chess and international chess pieces graph diameters”, which delves into the complexities of generalizing chess piece movements on k-dimensional chessboards. This paper provides a comprehensive perspective on k-pieces (i.e., chess pieces operating on k-dimensional n×n×⋯×n chessboards in Euclidean space). To round out your understanding, consider “On the existence of Hamiltonian cycles in hypercubes”, a shorter paper that expands the Euclidean knight’s tour concept to all the others fairy chess leapers, where the knight represents the mere (1, 2)-leaper and the paper considers each (a, b)-leaper for any pair of positive integers a≥0 and b>1.

These papers provide a solid foundation while highlighting many open problems and research directions for anyone eager to explore or expand on these findings.

Jacobsen: Thank you for the opportunity and your time, Marco.

Discussion

The interview between Scott Douglas Jacobsen and Marco Ripà offers an in-depth exploration of Ripà’s pioneering work in discrete mathematics, particularly focusing on integer tetration, minimal covering paths in high-dimensional grids, and advanced number theory. As an independent researcher and autodidact, Ripà has made significant contributions without formal academic affiliations, demonstrating exceptional creativity and analytical prowess. A central theme of the discussion is Ripà’s fascination with iterated exponentiation and the stabilization of rightmost digits in power towers. His work on congruence speed and minimal covering paths highlights his commitment to uncovering fundamental principles that address complex combinatorial problems. Ripà’s innovative approaches to classical puzzles, such as the Nine Dots Problem, and his extensions into higher dimensions showcase his ability to blend creativity with rigorous mathematical analysis.

Additionally, Ripà challenges existing theorems by providing constructive counterexamples in the realm of knight’s tours on higher-dimensional chessboards, illustrating his capacity to question and refine established mathematical concepts. The interview also touches upon the potential applications of his research in fields like cryptography and Ramsey theory, indicating a forward-thinking approach that seeks to bridge pure mathematics with practical applications. Ripà’s prolific contributions to the Online Encyclopedia of Integer Sequences (OEIS) and his efforts to engage the mathematical community through challenges and educational resources further emphasize his dedication to fostering mathematical curiosity and problem-solving skills. Overall, the discussion underscores Marco Ripà’s role as a visionary in discrete mathematics, whose independent research pushes the boundaries of combinatorics and number theory.

Methods

The interview with Marco Ripà from January 16, 2025, using a semi-structured format to allow for a comprehensive and flexible dialogue. Scott Douglas Jacobsen, the interviewer, prepared a series of targeted questions that covered various aspects of Ripà’s research, personal motivations, and future aspirations. The conversation took place virtually.

Peer-Reviewed Publications

Ripà, M., On the relation between perfect powers and tetration frozen digits, Journal of AppliedMath (eISSN: 2972-4805), Vol. 2 (2024), No. 5, 1771.

Ripà, M., Proving the existence of Euclidean knight’s tours on n × n × ⸱⸱⸱ × n chessboard for n < 4, Notes on Number Theory and Discrete Mathematics (ISSN: 1310-5132), Vol. 30 (2024), No. 1, pp. 20–33.

Ripà, M. and Onnis, L., Number of stable digits of any integer tetration, Notes on Number Theory and Discrete Mathematics (ISSN: 1310-5132), Vol. 28 (2022), No. 3, pp. 441–457.

Ripà, M., The congruence speed formula, Notes on Number Theory and Discrete Mathematics (ISSN: 1310-5132), Vol. 27 (2021), No. 4, pp. 43–61.

Ripà, M., General uncrossing covering paths inside the axis-aligned bounding box, Journal of Fundamental Mathematics and Applications (ISSN: 2621-6019), Vol. 4 (2021), No. 2, pp. 154–166.

Ripà, M., Reducing the Clockwise-Algorithm to k length classes, Journal of Fundamental Mathematics and Applications(ISSN: 2621-6019), Vol. 4 (2021), No. 1, pp. 61–68.

Ripà, M., Solving the 106 years old 3^k problem with the clockwise-algorithm, Journal of Fundamental Mathematics and Applications (ISSN: 2621-6019), Vol. 3 (2020), No. 2, pp. 84–97.

Ripà, M., On the constant congruence speed of tetration, Notes on Number Theory and Discrete Mathematics (ISSN:1310-5132), Vol. 26 (2020), No. 3, pp. 245-260.

Ripà, M., Covering paths in hypercubes: Conjecture about link length bounded from below, International Journal of Mathematical Archive (IJMA) (ISSN: 2229-5046), Vol. 10 (2019), No. 8, pp. 36–38.

Ripà, M., The 3 × 3 × ⋅⋅⋅ × 3 Points Problem solution, Notes on Number Theory and Discrete Mathematics (ISSN: 1310-5132), Vol. 25 (2019), No. 2, pp. 68–75.

Ripà, M., The n × n × n Points Problem optimal solution, Notes on Number Theory and Discrete Mathematics (ISSN:1310-5132), Vol. 22 (2016), No. 2, pp. 36–43.

Ripà, M., Tessaro, G., and Forti, A., Quanti numeri primi in 100 interi consecutivi?, Matematicamente.it Magazine (ISSN:2035-0449), Vol. 25 (2015), pp. 37–40.

Ripà, M., The Rectangular Spiral or the n_1 × n_2 × ⋅⋅⋅ × n_k Points Problem, Notes on Number Theory and Discrete Mathematics (ISSN: 1310-5132), Vol. 20 (2014), No. 1, pp. 59–71.
[A short and up-to-date version of the original NNTDM paper has been accepted by Optimization Online in August 2024, and is available at: https://optimization-online.org/?p=27442].

Ripà, M. and Morelli, G., Retro-analytical Reasoning IQ Tests for the High Range, Educational Research (ISSN: 2141-5161), Vol. 4 (2013), No. 4, pp. 309–320.

Ripà, M. and Dalmasso, E., Patterns Related to the Smarandache Circular Sequence Primality Problem, Notes on Number Theory and Discrete Mathematics (ISSN: 1310-5132), Vol. 18 (2012), No. 1, pp. 29–48.

Ripà, M., Identifying gifted children and dyslexia early diagnosis: risk of cheating on IQ tests, Conference Proceeding, 12th Asia-Pacific Conference on Giftedness, Dubai, July 17, 2012.

Ripà, M., Differentiating features of gifted children and dealing with IQ societies, Conference Proceeding, 12th Asia-Pacific Conference on Giftedness, Dubai, July 15, 2012.

Other Publications, Preprints, and Informative Articles

Jacobsen S.D. and Ripà, M., An Interview with Marco Ripà (Part Three), In-Sight Publishing
(ISSN 2369-6885), Vol. 10.A, May 2016.

Jacobsen S.D. and Ripà, M., An Interview with Marco Ripà (Part Two), In-Sight Publishing
(ISSN 2369-6885), Vol. 10.A, Feb. 2016.

Jacobsen S.D. and Ripà, M., An Interview with Marco Ripà (Part One), In-Sight Publishing
(ISSN 2369-6885), Vol. 10.A, Jan. 2016.

Ripà, M., Five Hard Problems with a Simple Solution, ResearchGate, January 2025. Available at:https://www.researchgate.net/publication/387704984_Five_Hard_Problems_with_a_Simple_Solution.

Ripà, M., Twelve Python Programs to Help Readers Test Peculiar Properties of Integer Tetration, Zenodo, December 2024. Available at: https://zenodo.org/records/14544793.

Ripà, M., Graham’s number stable digits: an exact solution, arXiv [math.GM], October 2024. Available at:https://arxiv.org/abs/2411.00015.

Di Pietro, G. and Ripà, M., On the existence of Hamiltonian cycles in hypercubes, arXiv [math.CO], September 2024. Available at: https://arxiv.org/abs/2409.03073.

Di Pietro, G. and Ripà, M., Euclidean Tours in Fairy Chess, arXiv [math.GM], June 2024. Available at: https://arxiv.org/abs/2407.07903.

Ripà, M., General conjecture on the optimal covering trails for any k-dimensional cubic lattice, IQ Nexus Journal, Vol. 16, No. 1, March 2024, pp. 55–61. Available at: https://iqnexus.org/2024/02/28/iqnj-v16-n1/.

Ripà, M., Congruence speed of tetration bases ending with 0, arXiv [math.NT], February 2024. Available at:https://arxiv.org/abs/2402.07929.

Ripà, M., A necessary and sufficient condition for the existence of Euclidean knight’s tours on 2 × 2 × ⋅⋅⋅ × 2 chessboards, ResearchGate, February 2024. Available at: https://www.researchgate.net/publication/378525989_A_necessary_and_sufficient_condition_for_the_existence_of_Euclidean_knight%27s_tours_on_2_2_2_chessboards.

Ripà, M., Metric spaces in chess and international chess pieces graph diameters, arXiv [math.HO], November 2023. Available at: https://arxiv.org/abs/2311.00016.

Rinaldi R. and Ripà, M., Optimal cycles enclosing all the nodes of a k-dimensional hypercube, arXiv [math.CO], December 2022. Available at: https://arxiv.org/abs/2212.11216.

Ripà, M., Shortest polygonal chains covering each planar square grid, arXiv [math.CO], July 2022. Available at: https://arxiv.org/abs/2207.08708.

Ripà, M., Minimum-Link Covering Trails for any Hypercubic Lattice, arXiv [math.GM], June 2022. Available at:https://arxiv.org/abs/2208.01699.

Ripà, M., On some open problems concerning perfect powers, IQ Nexus Journal, Vol. 14, No. 2, June 2022, pp. 41–48. Available at: https://iqnexus.org/2024/02/13/iqnj-v14-n2/.

Ripà, M., Very unbalanced Chess Positions, In-Sight: Independent Interview-Based Journal (ISSN: 2369-6885), Vol. 22.B, April 2020.

Ripà, M., Solving the n_1 × n_2 × n_3 Points Problem for n_3 < 6, In-Sight: Independent Interview-Based Journal (ISSN: 2369-6885), Vol. 22.B, Feb. 2020. [An up-to-date version of the original paper has been accepted by Optimization Online in June 2022, Updated in December 2024, and is available at: https://optimization-online.org/2022/06/8958/].

Ripà, M., Ricorsività delle cifre di particolari classi di interi, Rudi Mathematici – Bookshelf, RMBSH-027.

Ripà, M., Una curiosa proprietà, Rudi Mathematici – Bookshelf, RMBSH-026.

Ripà, M., Divisibilità per 3 degli elementi di alcune sequenze numeriche, Rudi Mathematici – Bookshelf, RMBSH-020.

Ripà, M., 2048 game: massimo punteggio, Matematicamente.it – Giochi e gare – Gioca con la matematica, No. 1, June 2014.

Ripà, M., Strani calcoli ispirati dal racconto di J. L. Borges ‘La biblioteca di Babele’, Matematicamente.it – Cultura –Matematica curiosa, No. 7, Nov. 2010.

Published Books

1729 il Numero di Mr. 17-29, Edizioni Eracle/Narrativa, Napoli, 2014 (ISBN: 978-8867430574).

Retro-analytical Reasoning IQ Tests for the High Range, LAP LAMBERT Academic Publishing, Saarbrücken, 2013 (ISBN: 978-3659437649).

Congetture su Interrogativi Inediti: tra Speculazioni, Voli Pindarici e Riflessioni Spicciole, Narcissus, Ebook, 2012 (ISBN: 978-8863699463).

La Strana Coda della Serie n^n^…^n, UNI Service, Trento, 2011 (ISBN: 978-8861787896).

Selected Public Debates and Prizes

Credited as the coauthor of the first dynamic spatial IQ tests, equally normed and automatically generated by software, in Scott Jacobsen’s article “On High-Range Test Construction 26: Marco Ripà and Roberto Enea, DynamIQ”, published online in November 2024 (available at: https://in-sightpublishing.com/2024/11/22/high-range-26/).

Online speaker at the 10th International Aegean Congress on Innovation Technologies & Engineering, October 5–7, 2024, Izmir, Turkey, presenting “Graham’s number stable digits: an exact result”.

In August 2024, Scott Douglas Jacobsen included “An Interview with Marco Ripà” (Part One, Two, and Three) in Some Smart People: Lives and Views 1, In-Sight Publishing, Fort Langley, British Columbia, Canada, pp. 251–269 (available at: https://in-sightpublishing.com/wp-content/uploads/2024/08/some-smart-people-lives-and-views-1.pdf).

Guest speaker at the national convention Gifted Education e Inclusione: miti, mode e misconcezioni, February 7, 2019, Lecce, Italy, discussing his school years as a gifted child.

In January 2017 participated in the debate on giftedness with R. Rosner and S. Jacobsen Ask A Genius (or Two) 68 –Conversation on Genius (5) (available at: https://rickrosner.org/2017/01/24/ask-a-genius-or-two-68-conversation-on-genius-5/).

In January 2016 was interviewed as GOTY winner by Scott D. Jacobsen for the In-Sight Publishing.

In June 2015, was interviewed about WIQF by the Korean TV ‘SBS’, The Genius Scouters.

In March 2015, he presented his latest novel at the public library F. Basaglia in Rome.

In March 2014 participated in the prime-time TV quiz Avanti un Altro!

Elected as “Genius of the Year 2014 – Europe” by members of the World Genius Directory and by the World Intelligence Network Executive Committee, mainly for his fundamental contribution to the creation of the first dynamic IQ tests (ENNDT and ENSDT).

In April 2013, he was hosted by the television program Romanzo Familiare as a “former gifted child” talking about acceleration, curriculum enrichment, and curricular compacting.

Guest speaker at the 12th Asia-Pacific Conference on Giftedness, July 14–18, 2012, Dubai, U.A.E., presenting two papers entitled “Identifying Gifted Children and Dyslexia Early Diagnosis: Risk of Cheating on IQ Tests” and “Differentiating Features of Gifted Children and Dealing with High IQ Societies”.

Data Availability

No datasets were generated or analyzed during the current article. All interview content remains the intellectual property of the interviewer and interviewee.

References

(No external academic sources were cited for this interview.)

Journal & Article Details

Publisher: In-Sight Publishing
Publisher Founding: March 1, 2014
Web Domain: http://www.in-sightpublishing.com
Location: Fort Langley, Township of Langley, British Columbia, Canada
Journal: In-Sight: Interviews
Journal Founding: August 2, 2012
Frequency: Four Times Per Year
Review Status: Non-Peer-Reviewed
Access: Electronic/Digital & Open Access
Fees: None (Free)
Volume Numbering: 13
Issue Numbering: 2
Section: A
Theme Type: Idea
Theme Premise: “Outliers and Outsiders”
Theme Part: 33
Formal Sub-Theme: None
Individual Publication Date: January 22, 2025
Issue Publication Date: April 1, 2025
Author(s): Scott Douglas Jacobsen
Word Count: 4,818
Image Credits: Photo by wu yi on Unsplash
ISSN (International Standard Serial Number): 2369-6885

Acknowledgements

The author thanks Marco Ripà for his time and willingness to participate in this interview.

Author Contributions

S.D.J. conceived and conducted the interview and prepared the manuscript.

Competing Interests

The author declares no competing interests.

License & Copyright

In-Sight Publishing by Scott Douglas Jacobsen is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
© Scott Douglas Jacobsen and In-Sight Publishing 2012–Present.

Unauthorized use or duplication of material without express permission from Scott Douglas Jacobsen is strictly prohibited. Excerpts and links must use full credit to Scott Douglas Jacobsen and In-Sight Publishing with direction to the original content.

Supplementary Information

Below are various citation formats for Breaking Bread with Marco Ripà on Iterated Exponentiation and the Nine Dots Problem (4).

American Medical Association (AMA 11th Edition)
Jacobsen S. Breaking Bread with Marco Ripà on Iterated Exponentiation and the Nine Dots Problem (4). January 2025;13(2). http://www.in-sightpublishing.com/ripà-4

American Psychological Association (APA 7th Edition)
Jacobsen, S. (2025, January 22). Breaking Bread with Marco Ripà on Iterated Exponentiation and the Nine Dots Problem (4). In-Sight Publishing. 13(2).

Brazilian National Standards (ABNT)
JACOBSEN, S. Breaking Bread with Marco Ripà on Iterated Exponentiation and the Nine Dots Problem (4). In-Sight: Interviews, Fort Langley, v. 13, n. 2, 2025.

Chicago/Turabian, Author-Date (17th Edition)
Jacobsen, Scott. 2025. “Breaking Bread with Marco Ripà on Iterated Exponentiation and the Nine Dots Problem (4).” In-Sight: Interviews 13 (2). http://www.in-sightpublishing.com/ripà-4.

Chicago/Turabian, Notes & Bibliography (17th Edition)
Jacobsen, S. “Breaking Bread with Marco Ripà on Iterated Exponentiation and the Nine Dots Problem (4).” In-Sight: Interviews 13, no. 2 (January 2025). http://www.in-sightpublishing.com/ripà-4.

Harvard
Jacobsen, S. (2025) ‘Breaking Bread with Marco Ripà on Iterated Exponentiation and the Nine Dots Problem (4)’, In-Sight: Interviews, 13(2). http://www.in-sightpublishing.com/ripà-4.

Harvard (Australian)
Jacobsen, S 2025, ‘Breaking Bread with Marco Ripà on Iterated Exponentiation and the Nine Dots Problem (4)’, In-Sight: Interviews, vol. 13, no. 2, http://www.in-sightpublishing.com/ripà-4.

Modern Language Association (MLA, 9th Edition)
Jacobsen, Scott. “Breaking Bread with Marco Ripà on Iterated Exponentiation and the Nine Dots Problem (4).” In-Sight: Interviews, vol. 13, no. 2, 2025, http://www.in-sightpublishing.com/ripà-4.

Vancouver/ICMJE
Jacobsen S. Breaking Bread with Marco Ripà on Iterated Exponentiation and the Nine Dots Problem (4) [Internet]. 2025 Jan;13(2). Available from: http://www.in-sightpublishing.com/ripà-4

Note on Formatting

This layout follows an adapted Nature research-article structure, tailored for an interview format. Instead of Methods, Results, and Discussion, we present Interview transcripts and a concluding Discussion. This design helps maintain scholarly rigor while accommodating narrative content.

#Combinatorics #CongruenceSpeed #Cryptography #DiscreteMathematics #GraphTheory #HighDimensionalGrids #IntegerTetration #KnightSTour #MinimalCoveringPaths #NineDotsPuzzle #NumberTheory #OEIS #PerfectPowers #RamseyTheory

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