mathematical-recreations   297
[1710.07145] Reaching a Target in the Plane with no Information
A mobile agent has to reach a target in the Euclidean plane. Both the agent and the target are modeled as points. In the beginning, the agent is at distance at most D>0 from the target. Reaching the target means that the agent gets at a {\em sensing distance} at most r>0 from it. The agent has a measure of length and a compass. We consider two scenarios: in the {\em static} scenario the target is inert, and in the {\em dynamic} scenario it may move arbitrarily at any (possibly varying) speed bounded by v. The agent has no information about the parameters of the problem, in particular it does not know D, r or v. The goal is to reach the target at lowest possible cost, measured by the total length of the trajectory of the agent.
Our main result is establishing the minimum cost (up to multiplicative constants) of reaching the target under both scenarios, and providing the optimal algorithm for the agent. For the static scenario the minimum cost is Θ((logD+log1r)D2/r), and for the dynamic scenario it is Θ((logM+log1r)M2/r), where M=max(D,v). Under the latter scenario, the speed of the agent in our algorithm grows exponentially with time, and we prove that for an agent whose speed grows only polynomially with time, this cost is impossible to achieve.
agent-based  optimization  mathematical-recreations  planning  algorithms  nudge-targets  consider:looking-to-see
3 days ago by Vaguery
Chopping a square into 7 triangles with almost the same area It's impossible...
It's impossible to divide a square into an odd number of triangles with the same area. This amazing result was proved by Paul Monsky in 1970, but I only heard about it today, from +David Eppstein, here on G+.

So it's impossible. But you can still try your best! These are the two best tries with 7 triangles and at most 8 vertices. These were discovered last year by Jean-Philippe Labbé, Günter Rote, and Günter Ziegler. They used a lot of clever math and programming to do this
mathematical-recreations  plane-geometry  nudge-targets  consider:looking-to-see
3 days ago by Vaguery
The problem with problems
I encountered this particular mathematical problem (by Peter Lijedahl) earlier this year at a meeting designed to provoke discussion, and expose elements, of ‘mathematical thinking’. The phrase ‘mathematical thinking’ is one that is used increasingly as something that we as educators and designers want to encourage, deepen, and develop in students - but it is very hard to quantify and observe. As we become more experienced problem solvers (and problem selectors!) we may forget what it’s like to be faced with a completely new mathematical problem and therefore the essential elements of early mathematical thinking which may be noticeable in our students.

So: it’s worth taking the time to examine our own thinking when we encounter new problems, to perform some forensic metacognition. If you can, spend five minutes reacting to the problem and a further five reflecting on your reaction.
10 days ago by Vaguery
Welcome to GF(4) – The Math Citadel
Notice that this solution is not the same as when we lived in our comfortable world of real numbers. The equations were the same, the numbers involved were the same, but we changed what addition and multiplication did by moving to a new field called GF(4). The purpose of this exercise was to get used to the idea of arithmetic in a new space, and to see what an example of a Galois field looks like. Explaining how to generate these Galois fields in general and defining their addition and multiplication tables will get a bit involved; we’ll tackle these soon. For now, it’s important just to let go of our tightly held arithmetic notions that are really special properties of real numbers. Systems of equations can yield different solutions when we move to a new world.
mathematics  heuristics  mathematical-recreations  to-write-about  group-theory  rather-interesting
22 days ago by Vaguery
Self-Replicating Functions
There are many exciting directions to explore from here, but I’m going to take a break now, considering this as a nice initial step into the realm of self-replicating functions.
23 days ago by Vaguery
Math-Frolic!: Playing Games
From the inimitable John Conway:
“You get surreal numbers by playing games. I used to feel guilty in Cambridge that I spent all day playing games, while I was supposed to be doing mathematics. Then, when I discovered surreal numbers, I realized that playing games IS mathematics.”
mathematical-recreations  mathematics  the-point  quotes
25 days ago by Vaguery
Mathematics must be creative, else it ain’t mathematics
When people ask, often cynically, what creativity looks like, it is surely this: the ability to join seemingly disparate ideas to form new expressions of thought and emotion.
By this definition, mathematics must be considered a creative pursuit. The mathematical world is governed by patterns and symmetries, some of them known and most of them awaiting our discovery.
There are no topics in mathematics; only artificial barriers that we have erected to help organise the curriculum. At school, we study topics in discrete chunks and come to understand them as separate islands of knowledge. Yet the most powerful and interesting mathematics arises when we cut through these barriers.
creativity  mathematics  mathematical-recreations  to-write-about  essay  system-of-professions
6 weeks ago by Vaguery
Game & Puzzle Design
Game & Puzzle Design is a peer-reviewed print journal publishing high quality work on all aspects of game and puzzle design. The journal aims to bring together designers and researchers from a variety of backgrounds, to foster a deeper understanding of games and facilitate the creation of new high quality games and puzzles. We are particularly interested in the intersection between traditional and digital game design, and the points at which these disciplines converge and diverge.

Submissions may pertain to any type of game or puzzle – abstract, physical, printed, digital, etc. – but should focus on underlying mechanics or gameplay rather than visual design. The emphasis will be on traditional games and puzzles, although submissions on digital games (other than major AAA video games) are also welcome, especially where links are drawn between traditional and digital design approaches.
7 weeks ago by Vaguery
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
′′′
.
9 weeks 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.
10 weeks 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
10 weeks 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.
10 weeks 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
10 weeks 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.
10 weeks 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
10 weeks 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
10 weeks 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
10 weeks 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
11 weeks ago by Vaguery

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