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In the decimal numeral system, we prove that the well-known Graham's number, $G := \! ^{n}3$ (i.e., $3^{3^{\cdot^{\cdot^{\cdot^{3}}}}} n$-times), and any base $3$ tetration whose hyperexponent is larger than $n$ share the same $slog_3(G) - 1$ rightmost digits (where $slog$ indicates the integer super-logarithm). This is an exact result since the $slog_3(G)$-th rightmost digit of $G$ is different from the $slog_3(G)$-th rightmost digit of $^{n+1}3$. Furthermore, we show that the $slog_3(^{n}3)$-th least significant digit of the difference between Graham's number and any base $3$ tetration whose integer hyperexponent is beyond $n$ is $4$., Comment: 9 pages, 1 figure

For each pair of positive integers $(a,b)$ such that $a \geq 0$ and $b > 1$, the present paper provides a necessary and sufficient condition for the existence of Hamiltonian cycles visiting all the vertices of any $k$-dimensional grid $\{0,1\}^k \subset \mathbb{R}^k$ and whose associated Euclidean distance is equal to $\sqrt{a^2+b^2}$. Our solution extends previously stated results in fairy chess on the existence of closed Euclidean $(a,b)$-leapers tours for $2 \times 2 \times \cdots \times 2$ chessboards, where the (Euclidean) knight identifies the $(1,2)$-leaper., Comment: 6 pages. Names of Lemma 2.3 and Corollary 2.5 fixed

A generalization of Rip\`a's square spiral solution for the $n \times n \times \cdots \times n$ Points Upper Bound Problem. Additionally, we provide a non-trivial lower bound for the $k$-dimensional $n_1 \times n_2 \times \cdots \times n_k$ Points Problem. In this way, we can build a range in which, with certainty, all the best possible solutions to the problem we are considering will fall. Finally, we give a few characteristic numerical examples in order to appreciate the fineness of the result arising from the particular approach we have chosen., Comment: 9 pages, 3 figures. This is a short version of the original paper, published in NNTDM, for the reasons explained at page 9

The present paper aims to extend the knight's tour problem for $k$-dimensional grids of the form $\{0,1\}^k$ to other fairy chess leapers. Accordingly, we constructively show the existence of closed tours in $2 \times 2 \times \cdots \times 2$ ($k$ times) chessboards concerning the wazir, the threeleaper, and the zebra, for all $k \geq 15$. Our result considers the three above-mentioned leapers and replicates for each of them the recent discovery of Euclidean knight's tours for the same set of $2 \times 2 \times \cdots \times 2$ grids, opening a new research path on the topic by studying different fairy chess leapers that perform jumps of fixed Euclidean length on given regular grids, visiting all their vertices exactly once before coming back to the starting one., Comment: 9 pages

For every non-negative integer $a$ and positive integer $b$, the congruence speed of the tetration $^{b}a$ is the difference between the number of the rightmost digits of $^{b}a$ that are the same as those of $^{b+1}a$ and the number of the rightmost digits of $^{b-1}a$ that are the same as those of $^{b}a$. In the decimal numeral system, if the given base $a$ is not a multiple of $10$, as $b:=b(a)$ becomes sufficiently large, we know that the value of the congruence speed does not depend on $b$ anymore, otherwise the number of the new rightmost zeros of $^{b}a$ drastically increases for any unit increment of $b$ and, for this reason, we have not previously described the congruence speed of $a$ when it is a multiple of $10$. This short note fills the gap by giving the formula for the congruence speed of the mentioned values of $a$ at any given height of the hyperexponent., Comment: 3 pages

Given the finite set of $n_1 \cdot n_2 \cdot \ldots \cdot n_k$ points $G_{n_1,n_2,\ldots,n_k} \subset \mathbb{R}^k$ such that $n_k \geq \cdots \geq n_2 \geq n_1 \in \mathbb{Z}^+$, we introduce a new algorithm, called M$\Lambda$I, which returns an uncrossing covering path inside the minimum axis-aligned bounding box $[0,n_1-1] \times [0,n_2-1] \times \cdots \times [0,n_k-1]$, consisting of $3 \cdot \prod_{i=1}^{k-1} n_i-2$ links of prescribed length $n_k-1$ units. Thus, for any $n_k \geq 3$, the link length of the covering path provided by our M$\Lambda$I-algorithm is smaller than the cardinality of the set $G_{n_1,n_2,\ldots,n_k}$. Furthermore, assuming $k>2$, we present an uncrossing covering path for $G_{3,3,\ldots,3}$, consisting of $20 \cdot 3^{k-3}-2$ straight-line edges that are $2$ units long each, which is constrained by the axis-aligned bounding box $\left[0,4-\sqrt{3}\right] \times \left[0,4-\sqrt{3}\right] \times [0, 2]^{k-2}$., Comment: 14 pages, 9 figures

This paper aims to study the graph radii and diameters induced by the $k$-dimensional versions of the well-known six international chess pieces on every finite $\{n \times n \times \dots \times n\} \subseteq \mathbb{Z}^k$ lattice since they originate as many interesting metric spaces for any proper pair $(n,k)$. For this purpose, we finally discuss a mathematically consistent generalization of all the planar FIDE chess pieces to an appropriate $k$-dimensional environment, finding (for any $k \in \mathbb{Z}^+$) the exact values of the graph radii and diameters of the $k$-rook, $k$-king, $k$-bishop, and the corresponding values for the $3$-queen, $3$-knight, and $3$-pawn. We also provide tight bounds for the graph radii and diameters of the $k$-queen, $k$-knight, and $k$-pawn, holding for any $k \geq 4$., Comment: 35 pages, 34 figures. References and a figure added; some typos corrected; improvements on grammar and style

The Knight's Tour problem consists of finding a Hamiltonian path for the knight on a given set of points so that the knight can visit exactly once every vertex of the mentioned set. In the present paper, we provide a $5$-dimensional alternative to the well-known statement that it is not ever possible for a knight to visit once every vertex of $C(3,k) := \{0,1,2\}^k$ by performing a sequence of $3^k-1$ jumps of standard length, since the most accurate answer to the original question actually depends on which mathematical assumptions we are making at the beginning of the game, when we decide to extend a planar chess piece to the third dimension and above. Our counterintuitive outcome follows from the observation that we can alternatively define a $2$D knight as a piece that moves from one square to another on the chessboard by covering a fixed Euclidean distance of $\sqrt{5}$ so that also the statement of Theorem~3 in [Erde, J., Gol{\'e}nia, B., \& Gol{\'e}nia, S. (2012), The closed knight tour problem in higher dimensions, The Electronic Journal of Combinatorics, 19(4), \#P9] does not hold anymore for such a Euclidean knight, as long as a $2 \times 2 \times \cdots \times 2$ chessboard with at least $2^6$ cells is given. Moreover, we show a classical closed knight's tour on $C(3,4)-\{(1,1,1,1)\}$ whose arrival is at a distance of $2$ from $(1,1,1,1)$, and we finally construct closed Euclidean knight's tours on $\{0,1\}^k$ for each integer $k \geq 6$., Comment: 14 pages, 4 figures. Version 2: Theorem 4.1 has been improved by including the k = 6 case, Reference added, and other minor modifications

We solve the general problem of visiting all the $2^k$ nodes of a $k$-dimensional hypercube by using a polygonal chain that has minimum link-length, and we show that this optimal value is given by $h(2,k):=3 \cdot 2^{k-2}$ if and only if $k \in \mathbb{N}-\{0,1\}$. Furthermore, for any $k$ above one, we constructively prove that it is possible to visit once and only once all the aforementioned nodes, $H(2,k):=\{\{0,1\} \times \{0,1\} \times \dots \times \{0,1\}\} \subset \mathbb{R}^k$, with a cycle (i.e., a closed path) having only $3 \cdot 2^{k-2}$ links., Comment: 13 pages, 5 figures. Typos corrected; misprint in Section 3 fixed

In the present paper we provide a formula that allows to compute the number of stable digits of any integer tetration base $a\in\mathbb{N}_0$. The number of stable digits, at the given height of the power tower, indicates how many of the last digits of the (generic) tetration are frozen. Our formula is exact for every tetration base which is not coprime to $10$, although a maximum gap equal to $V(a)+1$ digits (where $V(a)$ denotes the constant congruence speed of $a$) can occur, in the worst-case scenario, between the upper and lower bound. In addition, for every $a>1$ which is not a multiple of $10$, we show that $V(a)$ corresponds to the $2$-adic or $5$-adic valuation of $a-1$ or $a+1$, or even to the $5$-adic order of $a^{2}+1$, depending on the congruence class of $a$ modulo $20$., Comment: 18 pages

We solve a few open problems related to a peculiar property of the integer tetration ${^{b}a}$, which is the constancy of its congruence speed for any sufficiently large $b=b(a)$. Assuming radix-$10$ (the well-known decimal numeral system), we provide an explicit formula for the congruence speed $V(a) \in \mathbb{N}_0$ of any $a \in \mathbb{N}-\{0\}$ that is not a multiple of $10$. In particular, for any given $n \in \mathbb{N}$, we prove to be true Rip\`a's conjecture on the smallest $a$ such that $V(a)=n$. Moreover, for any $a \neq 1 : a \not\equiv 0 \pmod {10}$, we show the existence of infinitely many prime numbers $p_j:=p_j(V(a))$ such that $V(p_j)=V(a)$., Comment: 19 pages, 2 tables

Given any $n \in \mathbb{Z}^{+}$, we constructively prove the existence of covering paths and circuits in the plane which are characterized by the same link length of the minimum-link covering trails for the two-dimensional grid $G_n^2 := \{0,1, \ldots, n-1\} \times \{0, 1, \ldots, n-1\}$. Furthermore, we introduce a general algorithm that returns a covering cycle of analogous link length for any even value of $n$. Finally, we provide the tight upper bound $n^2 - 3 + 5 \cdot \sqrt{2}$ units for the minimum total distance travelled to visit all the nodes of $G_n^2$ with a minimum-link trail (i.e., a trail with $2 \cdot n - 2$ edges if $n$ is above two)., Comment: 18 pages, 13 figures; introduction improved, Kato's conjecture reformulated, and appendix moved after references

In 1994, Kranakis et al. published a conjecture about the minimum link-length of every rectilinear covering path for the $k$-dimensional grid $P(n,k) := \{0,1, \dots, n-1\} \times \{0,1, \dots, n-1\} \times \cdots \times \{0,1, \dots, n-1\}$. In this paper we consider the general, NP-complete, Line-Cover problem, where the edges are not required to be axis-parallel, showing that the original Theorem 1 by Kranakis et al. no longer holds when the aforementioned constraint is disregarded. Furthermore, for any given $n$ above two, as $k$ approaches infinity, the link-length of any minimal (non-rectilinear) polygonal chain does not exceed Kranakis' conjectured value of $\frac{k}{k-1} \cdot n^{k-1}+O(n^{k-2})$ only if we introduce a multiplicative constant $c \geq 1.5$ for the lower order terms (e.g., if we select $n=3$ and assume that $c<1.5$, starting from a sufficiently large $k$, it is not possible to visit all the nodes of $P(n,k)$ with a trail of link-length $\frac{k}{k-1} \cdot n^{k-1}+c \cdot n^{k-2}$)., Comment: 5 pages, extending and substantially improving some previously published results in Int. Journal of Math. Arch., Vol. 10(8), 36-38 (2019); minor stylistic corrections, reference 8 fixed; Improving Definitions 1.2 and 1.4, revising notations and fixing Theorem 2.1 accordingly (a clarifying example has also been added in the second paragraph of the proof)

The starting point of our paper is Kashihara's open problem number $30$, concerning the sequence $A001292$ of the OEIS, asking how many terms are powers of integers. We confirm his last conjecture up to the $100128$-th term and provide a general theorem that rules out $4/9$ of the candidates. Moreover, we formulate a new, provocative, conjecture involving the OEIS sequence $A352991$ (which includes all the terms of $A001292$). Our risky conjecture states that all the perfect powers belonging to the sequence $A352991$ are perfect squares and they cannot be written as higher order perfect powers if the given term of $A352991$ is not equal to one. This challenging conjecture has been checked for any integer smaller than $10111121314151617181920212223456789$ and no counterexample has been found so far., Comment: 8 pages; typos fixed, minor style improvements, and Cross-referencing added