mathematical-recreations   289
CTK Insights » Blog Archive A Discovery of Hirotaka Ebisui And Thanos Kalogerakis - CTK Insights
Hirotaka Ebisui has found that in the case of the right triangle,
A

+
B

=
5
C

A

+B

=5C

. On informing me of that result, Thanos added an identity for the next layer
A

+
B

=
C

.
A
′′
+B
′′
=C
′′
.

That was exciting enough for me to investigate. I can happily and proudly report that, for the next layer,
A

+
B

=
5
C

A
′′′
+B
′′′
=5C
′′′
.
6 days ago by Vaguery
Minus Infinity |
Today we’ll talk about some paradoxical things, like the logarithm of zero, and the maximum element of a set of real numbers that doesn’t contain any real numbers at all. More importantly, we’ll see how mathematicians try to wrap their heads around such enigmas.

All today’s logarithms will be base ten logarithms; so the logarithm of 100 is 2 (because 100 is 102) and the logarithm of 1/1000 is −3 (because 1/1000 is 10−3)). The logarithm of 0 would have to be an x that satisfies the equation 10x = 0. Since there’s no such number, we could just say “log 0 is undefined” and walk away, with our consciences clear and our complacency unruffled.
9 days ago by Vaguery
Swine in a Line |
What I hope you take away from this essay is a sense that there’s an interestingly tangled relationship between games, numbers, and the devices we use for representing and manipulating numbers. The abacus was introduced as a way to represent and manipulate numbers for purposes of counting. On the other hand, we can play games with an abacus (the Swine in a Line games) that don’t immediately seem to be about numbers at all, but if we play with those games for long enough, we come to see that underlying the chaos of one configuration giving rise to another, and then another, there is a pattern, and this pattern is most naturally expressed in the language of number.
number-theory  mathematical-recreations  math  pedagogy  ludics  to-write-about  sandpiles  complexology
9 days ago by Vaguery
Prof. Engel’s Marvelously Improbable Machines |
When the path from a simple question to a simple answer leads you through swamps of computation, you can accept that some amount of tromping through swamps is unavoidable in math and in life, or you can think harder and try to find a different route. This is a story of someone who thought harder.

His name is Arthur Engel. Back in the 1970s this German mathematician was in Illinois, teaching probability theory and other topics to middle-school and high-school students. He taught kids in grades 7 and up how to answer questions like “If you roll a fair die, how long on average should you expect to wait until the die shows a three?” The questions are simple, and the answers also tend to be simple: whole numbers, or fractions with fairly small numerators and denominators. You can solve these problems using fraction arithmetic (in the simpler cases) or small systems of linear equations (for more complicated problems), and those are the methods that Engel taught his students up through the end of 1973.
9 days ago by Vaguery
[cs/0305016] The one-round Voronoi game replayed
We consider the one-round Voronoi game, where player one (White'', called Wilma'') places a set of n points in a rectangular area of aspect ratio r <=1, followed by the second player (Black'', called Barney''), who places the same number of points. Each player wins the fraction of the board closest to one of his points, and the goal is to win more than half of the total area. This problem has been studied by Cheong et al., who showed that for large enough n and r=1, Barney has a strategy that guarantees a fraction of 1/2+a, for some small fixed a.
We resolve a number of open problems raised by that paper. In particular, we give a precise characterization of the outcome of the game for optimal play: We show that Barney has a winning strategy for n>2 and r>sqrt{2}/n, and for n=2 and r>sqrt{3}/2. Wilma wins in all remaining cases, i.e., for n>=3 and r<=sqrt{2}/n, for n=2 and r<=sqrt{3}/2, and for n=1. We also discuss complexity aspects of the game on more general boards, by proving that for a polygon with holes, it is NP-hard to maximize the area Barney can win against a given set of points by Wilma.
mathematical-recreations  game-theory  rather-interesting  planning  consider:looking-to-see  nudge-targets  to-write-about  to-simulate
12 days ago by Vaguery
[1306.4884] Cannibal Animal Games: a new variant of Tic-Tac-Toe
This paper presents a new partial two-player game, called the \emph{cannibal animal game}, which is a variant of Tic-Tac-Toe. The game is played on the infinite grid, where in each round a player chooses and occupies free cells. The first player Alice can occupy a cell in each turn and wins if she occupies a set of cells, the union of a subset of which is a translated, reflected and/or rotated copy of a previously agreed upon polyomino P (called an \emph{animal}). The objective of the second player Bob is to prevent Alice from creating her animal by occupying in each round a translated, reflected and/or rotated copy of P. An animal is a \emph{cannibal} if Bob has a winning strategy, and a \emph{non-cannibal} otherwise. This paper presents some new tools, such as the \emph{bounding strategy} and the \emph{punching lemma}, to classify animals into cannibals or non-cannibals. We also show that the \emph{pairing strategy} works for this problem.
12 days ago by Vaguery
Knots, Links, & Learning | arbitrarilyclose
If any of this is compelling to you, I would love it if you wanted to join me in this investigation. I don’t want someone to show up and say, “here’s the final solution, dummy, it’s already all been solved,” because I’m sure that this is already a chapter in a textbook somewhere. I don’t care about that. I care about investigating for myself and with friends what’s happening. Feel free to share ideas below, and I would love it if you did some of your own drawings and joined in on the party.
learning-in-public  mathematical-recreations  looking-to-see  lovely
12 days ago by Vaguery
[1108.3615v1] Interactions between Digital Geometry and Combinatorics on Words
We review some recent results in digital geometry obtained by using a combinatorics on words approach to discrete geometry. Motivated on the one hand by the well-known theory of Sturmian words which model conveniently discrete lines in the plane, and on the other hand by the development of digital geometry, this study reveals strong links between the two fields. Discrete figures are identified with polyominoes encoded by words. The combinatorial tools lead to elegant descriptions of geometrical features and efficient algorithms. Among these, radix-trees are useful for efficiently detecting path intersection, Lyndon and Christoffel words appear as the main tools for describing digital convexity; equations on words allow to better understand tilings by translations.
combinatorics  representation  rather-interesting  strings  tiling  to-write-about  domino-tiling  mathematical-recreations
13 days ago by Vaguery
[1504.00212] New results on the stopping time behaviour of the Collatz 3x + 1 function
Let σn=⌊1+n⋅log23⌋. For the Collatz 3x + 1 function exists for each n∈ℕ a set of different residue classes (mod 2σn) of starting numbers s with finite stopping time σ(s)=σn. Let zn be the number of these residue classes for each n≥0 as listed in the OEIS as A100982. It is conjectured that for each n≥4 the value of zn is given by the formula
zn=(⌊5(n−2)3⌋n−2)−∑i=2n−1(⌊3(n−i)+δ2⌋n−i)⋅zi,
where δ∈ℤ assumes different values within the sum at intervals of 5 or 6 terms. This allows to create an iterative algorithm which generates zn for each n>12. This has been proved for each n≤10000. The number z10000 has 4527 digits.
number-theory  mathematical-recreations  rather-interesting  to-write-about  nudge-targets  consider:looking-to-see  feature-construction
14 days ago by Vaguery
[0705.4085] The Distance Geometry of Music
We demonstrate relationships between the classic Euclidean algorithm and many other fields of study, particularly in the context of music and distance geometry. Specifically, we show how the structure of the Euclidean algorithm defines a family of rhythms which encompass over forty timelines (\emph{ostinatos}) from traditional world music. We prove that these \emph{Euclidean rhythms} have the mathematical property that their onset patterns are distributed as evenly as possible: they maximize the sum of the Euclidean distances between all pairs of onsets, viewing onsets as points on a circle. Indeed, Euclidean rhythms are the unique rhythms that maximize this notion of \emph{evenness}. We also show that essentially all Euclidean rhythms are \emph{deep}: each distinct distance between onsets occurs with a unique multiplicity, and these multiplicies form an interval 1,2,...,k−1. Finally, we characterize all deep rhythms, showing that they form a subclass of generated rhythms, which in turn proves a useful property called shelling. All of our results for musical rhythms apply equally well to musical scales. In addition, many of the problems we explore are interesting in their own right as distance geometry problems on the circle; some of the same problems were explored by Erd\H{o}s in the plane.
music  mathematical-recreations  computational-geometry  rather-interesting  to-write-about  rhythm
21 days ago by Vaguery
[1211.2734] Fixed-Orientation Equilateral Triangle Matching of Point Sets
Given a point set P and a class  of geometric objects, G(P) is a geometric graph with vertex set P such that any two vertices p and q are adjacent if and only if there is some C∈ containing both p and q but no other points from P. We study G▽(P) graphs where ▽ is the class of downward equilateral triangles (ie. equilateral triangles with one of their sides parallel to the x-axis and the corner opposite to this side below that side). For point sets in general position, these graphs have been shown to be equivalent to half-Θ6 graphs and TD-Delaunay graphs.
The main result in our paper is that for point sets P in general position, G▽(P) always contains a matching of size at least ⌈n−23⌉ and this bound cannot be improved above ⌈n−13⌉.
We also give some structural properties of $G_{\davidsstar}(P)$ graphs, where $\davidsstar$ is the class which contains both upward and downward equilateral triangles. We show that for point sets in general position, the block cut point graph of $G_{\davidsstar}(P)$ is simply a path. Through the equivalence of $G_{\davidsstar}(P)$ graphs with Θ6 graphs, we also derive that any Θ6 graph can have at most 5n−11 edges, for point sets in general position.
21 days ago by Vaguery
[0705.0635] Moving Walkways, Escalators, and Elevators
We study a simple geometric model of transportation facility that consists of two points between which the travel speed is high. This elementary definition can model shuttle services, tunnels, bridges, teleportation devices, escalators or moving walkways. The travel time between a pair of points is defined as a time distance, in such a way that a customer uses the transportation facility only if it is helpful.
We give algorithms for finding the optimal location of such a transportation facility, where optimality is defined with respect to the maximum travel time between two points in a given set.
operations-research  rather-interesting  computational-geometry  performance-measure  nudge-targets  mathematical-recreations  to-write-about  consider:looking-to-see
21 days ago by Vaguery
[1702.01027] Random Triangles and Polygons in the Plane
We consider the problem of finding the probability that a random triangle is obtuse, which was first raised by Lewis Caroll. Our investigation leads us to a natural correspondence between plane polygons and the Grassmann manifold of 2-planes in real n-space proposed by Allen Knutson and Jean-Claude Hausmann. This correspondence defines a natural probability measure on plane polygons. In these terms, we answer Caroll's question. We then explore the Grassmannian geometry of planar quadrilaterals, providing an answer to Sylvester's four-point problem, and describing explicitly the moduli space of unordered quadrilaterals. All of this provides a concrete introduction to a family of metrics used in shape classification and computer vision.
probability-theory  open-problems  mathematical-recreations  geometry  rather-interesting  to-write-about
29 days ago by Vaguery
[1707.03894] The Generalized Nagell-Ljunggren Problem: Powers with Repetitive Representations
We consider a natural generalization of the Nagell-Ljunggren equation to the case where the qth power of an integer y, for q >= 2, has a base-b representation that consists of a length-l block of digits repeated n times, where n >= 2. Assuming the abc conjecture of Masser and Oesterl\'e, we completely characterize those triples (q, n, l) for which there are infinitely many solutions b. In all cases predicted by the abc conjecture, we are able (without any assumptions) to prove there are indeed infinitely many solutions.
number-theory  feature-construction  rather-interesting  mathematical-recreations  nudge-targets  consider:looking-to-see  consider:classification
4 weeks ago by Vaguery
Counting Your Chickens Before They’re Pecked | bit-player
It started with a brief story in the New York Times about Luke Robitaille, a 13-year-old student from Euless, Texas, who won the Raytheon Mathcounts National Competition by correctly answering the following question:
In a barn, 100 chicks sit peacefully in a circle. Suddenly, each chick randomly pecks the chick immediately to its left or right. What is the expected number of unpecked chicks?
mathematical-recreations  probability-theory  simulation  learning-in-public  rather-interesting  to-write-about
4 weeks ago by Vaguery
Math Notes - Futility Closet
Can a sum of such fractions produce any natural number? That’s conjectured — but so far unproven.
number-theory  continued-fractions  mathematical-recreations  open-questions  nudge-targets  consider:looking-to-see  consider:performance-measures
5 weeks ago by Vaguery
[1012.1128] Yet another aperiodic tile set
We present here an elementary construction of an aperiodic tile set. Although there already exist dozens of examples of aperiodic tile sets we believe this construction introduces an approach that is different enough to be interesting and that the whole construction and the proof of aperiodicity are hopefully simpler than most existing techniques.
tiling  aperiodic-tiling  construction  rather-interesting  mathematical-recreations  to-write-about  to-simulate  consider:expanding
5 weeks ago by Vaguery
New Shapes Solve Infinite Pool-Table Problem | Quanta Magazine
Two “rare jewels” have illuminated a mysterious multidimensional object that connects a huge variety of mathematical work.
dynamical-systems  geometry  rather-interesting  mathematical-recreations  nudge-targets  consider:looking-to-see  consider:feature-discovery
5 weeks ago by Vaguery

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