mathematical-recreations   342

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Quickly recognizing primes less than 100 | The Math Less Traveled
Recently, Mark Dominus wrote about trying to memorize all the prime numbers under . This is a cool idea, but it made me start thinking about alternative: instead of memorizing primes, could we memorize a procedure for determining whether a number under is prime or composite? And can we make it clever enough to be done relatively quickly? This does tie into my other recent posts about primality testing, but to be clear, it’s also quite different: I am not talking about a general method for determining primality, but the fastest method we can devise for mentally determining, by hook or by crook, whether a given number under is prime. Hopefully there are rules we can come up with which are valid for numbers less than —and thus make them easier to test—even though they aren’t valid for bigger numbers in general.
mathematical-recreations  number-theory  heuristics  nudge-targets  consider:looking-to-see  to-write-about 
3 days ago by Vaguery
The Kolakoski Sequence - Futility Closet
This is a fractal, a mathematical object that encodes its own representation. It was described by William Kolakoski in 1965, and before him by Rufus Oldenburger in 1939. University of Evansville mathematician Clark Kimberling is offering a reward of $200 for the solution to five problems associated with the sequence:

Is there a formula for the nth term?
If a string occurs in the sequence, must it occur again?
If a string occurs, must its reversal also occur?
If a string occurs, and all its 1s and 2s are swapped, must the new string occur?
Does the limiting frequency of 1s exist, and is it 1/2?

So far, no one has found the answers.
mathematical-recreations  self-reference  fractals  open-questions  nudge-targets  consider:looking-to-see 
10 days ago by Vaguery
Tiling with TriCurves
There are a number of ways one can define a tricurve, the one used here is to start with an arc of some angle, replicate two identical curves ard rotate each about some angle about the ends of the arc. The Tricurve is the enclosed area.
plane-geometry  tiling  rather-interesting  define-your-terms  representation  to-write-about  mathematical-recreations 
10 days ago by Vaguery
Tiling with One Arc-Sided Shape | Math ∞ Blog
A flat puzzle (tiling) with dozens or hundreds of identical pieces may sound a little dull and predictable. But what is the most interesting shape we can use, to get the most unusual designs and the most variety? To make it more visually interesting, let’s say we want a shape with no straight edges—only curves. The following guidelines should help us get started.
plane-geometry  representation  tiling  rather-interesting  mathematical-recreations  to-write-about 
10 days ago by Vaguery
Efficiency of repeated squaring | The Math Less Traveled
Claim: the binary algorithm is the most efficient way to build using only doubling and incrementing steps. That is, any other way to build by doubling and incrementing uses an equal or greater number of steps than the binary algorithm.
algorithms  mathematical-recreations  nudge-targets  consider:looking-to-see  to-write-about 
11 days ago by Vaguery
Inside “3-adica” – From Earth to the Stars
Why can’t there be a square root of 2, though? Surely we can at least get close? Can’t we just copy what we saw with 7, and find a number that is (A) an integer squared, and (B) differs from 2 by a (3-adically) “small” quantity that is divisible by a large power of 3?

The problem is, if you square any integer at all, its remainder after you divide by 3 will either be zero (when the number you squared was exactly divisible by 3), or 1, in every other case. If you then subtract 2, the result can’t possibly be divisible by even a single power of 3. So any number squared will lie at a 3-adic distance of at least 1 from 2.

This peculiar way of measuring the distance between numbers turns out to be much more than a game. Everything we’ve done with the number 3 can be generalized to any other prime number, p, to create what are known as the p-adic numbers. Working with this kind of number lets you combine different aspects of mathematics in a very fruitful manner, and even the famous proof by Andrew Wiles of Fermat’s Last Theorem employed techniques that used p-adic numbers.
mathematical-recreations  number-theory  representation  to-write-about 
14 days ago by Vaguery
MathsJam
MathsJam is a monthly opportunity for like-minded self-confessed maths enthusiasts to get together in a pub and share stuff they like. Puzzles, games, problems, or just anything they think is cool or interesting.

We meet on the second-to-last Tuesday of every month, from around 7pm in the evening, in locations around the world.

For more details of local events, choose your region from the menu, or visit the find a jam page.
mathematical-recreations  to-do 
29 days ago by Vaguery
Roto-Tiler – The Inner Frame
Today we look at a puzzle invented by Alan Schoen that he calls Roto-Tiler. He explained this to me a few years ago, and when I showed him notes I made for a class, he denied that this is the puzzle he described. I insist it is, and it is quite certainly not mine.
mathematical-recreations  puzzles  tiling  representation  rather-interesting  to-write-about  nudge-targets 
4 weeks ago by Vaguery
The Mathematical Beauty of the Game SET | The Aperiodical
It is my hope that this post has given you some insight into the deep and elegant complexities of the game of SET besides just trying to be the fastest SET-finder in a game. It is miraculous that such a simple game can have such beautiful and joyful connections to some advanced domains of mathematics, even with some open research questions. If interested, the reader is encouraged to explore some of these considerations or generalizations, either playing with finite geometry or doing some combinatorics. The SET community has also developed some other very interesting variations on SET and their connections to other geometries, such as projective geometry, known as ProSet.
mathematical-recreations  games  set-theory  representation  have-explored  nudge-targets  to-write-about 
7 weeks ago by Vaguery
SMT solutions | The Math Less Traveled
In my last post I described the general approach I used to draw orthogons using an SMT solver, but left some of the details as exercises. In this post I’ll explain the solutions I came up with.
geometry  programming  rather-interesting  representation  testing  to-write-about  mathematical-recreations  nudge-targets  consider:looking-to-see 
8 weeks ago by Vaguery
Figuring out when you can do a puzzle. – Occupy Math
This week’s Occupy Math looks at a type of puzzle where you want to fill a rectangle with a shape. We will be using the L-shaped 3-square polyomino, used to fill a 5×9 rectangle below, as our example shape. The goal is to figure out every possible size of rectangle that can be filled with this shape. If you are constructing puzzles for other people — e.g., your students — knowing which problems can be solved gives you an edge. The post will not only solve the problem for our example shape, but also give you tools for doing this for other shapes. The answers, and the tools, are at the bottom if you don’t feel like working through the reasoning.
mathematical-recreations  polyominoes  proof  rather-interesting  nudge-targets  consider:classification  consider:feature-discovery 
10 weeks ago by Vaguery
#QuarterTheCross Card Sort – Wonder in Mathematics
It is no secret that Quarter the Cross is one of my favourite tasks. I’ve written about it twice before: as a Day 1 activity and in connection with Fraction Talks. The original source is apparently T. Dekker & N. Querelle, 2002, Great Assessment Problems (www.fi.uu.nl/catch). It has proliferated in recent years, including with an active Twitter hashtag: #QuarterTheCross.
mathematical-recreations  nudge-targets  consider:novelty-search  innovation  to-write-about  learning-by-doing 
10 weeks ago by Vaguery
Custom Baking - Futility Closet
Is it possible to bake a cake that can be divided into four parts by a single straight cut?
mathematical-recreations  simplicity  nudge-targets  consider:looking-to-see  consider:novelty-search 
10 weeks ago by Vaguery
[1803.06610] Can You Pave the Plane Nicely with Identical Tiles
Every body knows that identical regular triangles or squares can tile the whole plane. Many people know that identical regular hexagons can tile the plane properly as well. In fact, even the bees know and use this fact! Is there any other convex domain which can tile the Euclidean plane? Yes, there is a long list of them! To find the list and to show the completeness of the list is a unique drama in mathematics, which has lasted for more than one century and the completeness of the list has been mistakenly announced not only once! Up to now, the list consists of triangles, quadrilaterals, three types of hexagons, and fifteen types of pentagons. In 2017, Michael Rao announced a computer proof for the completeness of the list. Meanwhile, Qi Yang and Chuanming Zong made a series of unexpected discoveries in multiple tilings in the Euclidean plane. For examples, besides parallelograms and centrally symmetric hexagons, there is no other convex domain which can form any two-, three- or four-fold translative tiling in the plane; there are only two types of octagons and one type of decagons which can form five-fold translative tilings.
tiling  mathematical-recreations  marjorie-rice  to-write-about  plane-geometry 
12 weeks ago by Vaguery
[1803.08530] A Coloring Book Approach to Finding Coordination Sequences
An elementary method is described for finding the coordination sequences for a tiling, based on coloring the underlying graph. We illustrate the method by first applying it to the two kinds of vertices (tetravalent and trivalent) in the Cairo (or dual-3^2.4.3.4) tiling. The coordination sequence for a tetravalent vertex turns out, surprisingly, to be 1, 4, 8 ,12, 16, ..., the same as for a vertex in the familiar square (or 4^4) tiling. We thought that such a simple fact should have a simple proof, and this article is the result. We also apply the method to obtain coordination sequences for the 3^2.4.3.4, 3.4.6.4, 4.8^2, 3.12^2, and 3^4.6 uniform tilings, as well as the snub-632 and bew tilings. In several cases the results provide proofs for previously conjectured formulas.
combinatorics  feature-construction  representation  rather-interesting  enumeration  to-write-about  mathematical-recreations  consider:random-graphs  consider:non-tilings 
july 2018 by Vaguery
Exploring the golden ratio Φ
This post is really just things I've learned from other people, but things that surprised me
mathematical-recreations  phi  geometry  to-write-about 
may 2018 by Vaguery
The Metonymy of Matrices - Scientific American Blog Network
As a tool, the matrix is so powerful that it is easy to forget that it is a representation of a function, not a function itself. A matrix truly is just the array of numbers, but I think in this context, most mathematicians are metonymists. (Metonymers? Metonymnistes?) We think of the matrix as the function itself, and it’s easy to lose sight of the fact that it's only notation. Matrices don’t even have to encode linear transformations. They are used in other contexts in mathematics, too, and restricting our definition to linear transformations can shortchange the other applications (though for my money, the value of the matrix as a way of representing linear transformations dwarfs any other use they have).
matrices  representation  functions  mathematical-recreations  to-write-about 
may 2018 by Vaguery
Orthogonal polygons | The Math Less Traveled
Quite a few commenters figured out what was going on, and mentioned several nice (equivalent) ways to think about it. Primarily, the idea is to draw all possible orthogonal polygons, that is, polygons with only right angles, organized by the total number of vertices. (So, for example, the picture above shows all orthogonal polygons with exactly ten vertices.) However, we have to be careful what we mean by the phrase “all possible”: there would be an infinite number of such polygons if we think about things like the precise lengths of edges. So we have to say when two polygons are considered the same, and when they are distinct. My rules are as follows:
mathematical-recreations  looking-to-see  geometry  polyominoes  to-write-about 
may 2018 by Vaguery

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