**mathematical-recreations**342

Quickly recognizing primes less than 100 | The Math Less Traveled

3 days ago by Vaguery

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

10 days ago by Vaguery

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
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.

10 days ago by Vaguery

Tiling with TriCurves

10 days ago by Vaguery

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

10 days ago by Vaguery

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

11 days ago by Vaguery

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

Patterns That Eventually Fail | Azimuth

11 days ago by Vaguery

Sometimes patterns can lead you astray.

mathematical-recreations
mathematics
patterns
rather-interesting
to-write-about
11 days ago by Vaguery

Inside “3-adica” – From Earth to the Stars

14 days ago by Vaguery

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
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.

14 days ago by Vaguery

MathsJam

29 days ago by Vaguery

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
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.

29 days ago by Vaguery

Roto-Tiler – The Inner Frame

4 weeks ago by Vaguery

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

7 weeks ago by Vaguery

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

8 weeks ago by Vaguery

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

10 weeks ago by Vaguery

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

10 weeks ago by Vaguery

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

10 weeks ago by Vaguery

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

12 weeks ago by Vaguery

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

july 2018 by Vaguery

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

may 2018 by Vaguery

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

[1602.06208v2] Every positive integer is a sum of three palindromes

may 2018 by Vaguery

For integer g≥5, we prove that any positive integer can be written as a sum of three palindromes in base g.

mathematical-recreations
number-theory
nudge-targets
consider:rediscovery
consider:representation
to-write-about
benchmarks
reveal-intent
may 2018 by Vaguery

The Metonymy of Matrices - Scientific American Blog Network

may 2018 by Vaguery

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

may 2018 by Vaguery

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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