38406501359372282063949 & all that: Monodromy of Fano Problems
Article by Sachi Hashimoto and Borys Kadets
In collections: Attention-grabbing titles, Combinatorics, Integerology
A Fano problem is an enumerative problem of counting $$r$$-dimensional linear subspaces on a complete intersection in $$\mathbb{P}^n$$ over a field of arbitrary characteristic, whenever the corresponding Fano scheme is finite. A...
URL: arxiv.org/abs/2002.04580v1
PDF: arxiv.org/pdf/2002.04580v1

Fractal Sequences
Web page by Clark Kimberling
In collections: Easily explained, Fun maths facts, Integerology, Puzzles
Fractal sequences have in common with the more familiar geometric fractals the property of self-containment. An example of a fractal sequence is 1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 7, 4, 8, 1, 9, 5, 10, 3, 11, 6, 12, 2, 13, 7, 14, 4, 15, 8, . . . If you delete the first occurrence of each positive integer,...
URL: faculty.evansville.edu/ck6/int

An aperiodic hexagonal tile
Article by Socolar, Joshua E. S. and Taylor, Joan M.
In collection: Geometry
We show that a single tile can fill space uniformly but not admit a periodic tiling. The space--filling tiling that can be built from copies of the tile has the structure of a union of honeycombs with lattice constants of $$2^n a$$, where $$a$$ sets the scale of the most dense lattice and $$n$$ takes...
URL: arxiv.org/abs/1003.4279
PDF: arxiv.org/pdf/1003.4279v2

Mechanical Computing Systems Using Only Links and Rotary Joints
Article by Ralph C. Merkle and Robert A. Freitas Jr. and Tad Hogg and Thomas E. Moore and Matthew S. Moses and James Ryley
In collections: Basically computer science, Things to make and do, Unusual computers
A new paradigm for mechanical computing is demonstrated that requires only two basic parts, links and rotary joints. These basic parts...
URL: arxiv.org/abs/1801.03534v1
PDF: arxiv.org/pdf/1801.03534v1

A knowledge-based approach of connect-four
Article by Allis, Victor
In collection: Games to play with friends
A Shannon C-type strategy program, VICTOR, is written for Connect-Four, based on nine strategic rules. Each of these rules is proven to be correct, implying that conclusions made by VICTOR are correct. Using VICTOR, strategic rules where found which can be used by Black to at least draw the game, on each 7 × (2n) board,...
URL: informatik.uni-trier.de/~ferna

Redefining the integral
Article by Derek Orr
In collection: Unusual arithmetic
In this paper, we discuss a similar functional to that of a standard integral. The main difference is in its definition: instead of taking a sum, we are taking a product. It turns out this new "star-integral" may be written in terms of the standard integral but it has many different (and similar) interesting properties compared...
URL: arxiv.org/abs/1805.01861v1
PDF: arxiv.org/pdf/1805.01861v1

Perimeter-minimizing pentagonal tilings
Article by Chung, Ping Ngai and Fernandez, Miguel and Shah, Niralee and Sordo Vieira, Luis and Wikner, Elena
In collections: Easily explained, Geometry, Things to make and do
We provide examples of perimeter-minimizing tilings of the plane by convex pentagons and examples of perimeter-minimizing tilings of certain small flat tori.
URL: msp.org/involve/2014/7-4/p02.x
PDF: msp.org/involve/2014/7-4/invol

Hilbert 13: Are there any genuine continuous multivariate real-valued functions?
Article by Morris, Sidney
In collections: About proof, Fun maths facts
This article begins with a provocative question: Are there any genuine continuous multivariate real-valued functions? This may seem to be a silly question, but it is in essence what David Hilbert asked as one of the 23 problems he posed at the second...
URL: ams.org/journals/bull/2021-58-
PDF: ams.org/bull/2021-58-01/S0273-

How do you fix an Oval Track Puzzle?
Article by David A. Nash and Sara Randall
In collections: Easily explained, Puzzles
The oval track group, $$OT_{n,k}$$, is the subgroup of the symmetric group, $$S_n$$, generated by the basic moves available in a generalized oval track puzzle with $$n$$ tiles and a turntable of size $$k$$. In this paper we completely describe the oval track group for all possible...
URL: arxiv.org/abs/1612.04476v3
PDF: arxiv.org/pdf/1612.04476v3

A combinatorial approach to sums of two squares and related problems
In collections: About proof, Fun maths facts
Heath-Brown [6] suggested a short proof of the two squares theorem, thereby simplifying ideas of Liouville. Zagier [15] suggested a particularly neat form of this, a "One sentence proof". It consists of two suitable involutions on the finite set of the solutions of p = x 2 +4yz in positive integers. A parity argument...

Quotients Homophones des Groupes Libres Homophonic Quotients of Free Groups
Article by Washington, Lawrence and Zagier, Don
In collections: Easily explained, The groups group
URL: people.mpim-bonn.mpg.de/zagier

Reversible quantum cellular automata
Article by Schumacher, B. and Werner, R. F.
In collection: Basically computer science
We define quantum cellular automata as infinite quantum lattice systems with discrete time dynamics, such that the time step commutes with lattice translations and has strictly finite propagation speed. In contrast to earlier definitions this allows us to give an explicit characterization of all local rules...
URL: arxiv.org/abs/quant-ph/0405174

Fibonacci Jigsaw Puzzle
Article by Akio Hizume
In collections: Puzzles, Easily explained, Fibonaccinalia
URL: starcage.org/fibonacci_puzzle/

Conway's Wizards
Article by Khovanova, Tanya
In collections: Puzzles, Easily explained
I present and discuss a puzzle about wizards invented by John H. Conway.
URL: arxiv.org/abs/1210.5460
PDF: arxiv.org/pdf/1210.5460v1

A categorical foundation for Bayesian probability
Article by Culbertson, Jared and Sturtz, Kirk
In collection: Probability and statistics
Given two measurable spaces $$H$$ and $$D$$ with countably generated $$\sigma$$-algebras, a prior probability measure $$P_H$$ on $$H$$ and a sampling distribution $$\mcS:H \rightarrow D$$, there is a corresponding inference map $$\mcI:D \rightarrow H$$ which is unique...
URL: arxiv.org/abs/1205.1488
PDF: arxiv.org/pdf/1205.1488v3

On the interval containing at least one prime number
Article by Jitsuro Nagura
In collections: Fun maths facts, Integerology
URL: projecteuclid.org/euclid.pja/1

Orange Peels and Fresnel Integrals
Article by Bartholdi, Laurent and Henriques, André G.
In collections: Attention-grabbing titles, Easily explained, Things to make and do, Food, Geometry, Fun maths facts
There are two standard ways of peeling an orange: either cut the skin along meridians, or cut it along a spiral. We consider here the second method, and study the shape of the spiral strip, when...
URL: arxiv.org/abs/1202.3033
PDF: arxiv.org/pdf/1202.3033v1

$$H$$-supermagic labelings for firecrackers, banana trees and flowers
Article by Rachel Wulan Nirmalasari Wijaya and Andrea Semaničová-Feňovčíková and Joe Ryan and Thomas Kalinowski
In collections: Attention-grabbing titles, Basically computer science, Food
A simple graph $$G=(V,E)$$ admits an $$H$$-covering if every edge in $$E$$ is contained in a subgraph $$H'=(V',E')$$ of $$G$$ which is isomorphic to...
URL: arxiv.org/abs/1607.07911v2
PDF: arxiv.org/pdf/1607.07911v2

Indigenous perspectives in maths: Understanding Gurruṯu
Article by Chris Matthews
In collections: Easily explained, Fun maths facts
Discusses Yolŋu mathematics and the interconnected relationships of Gurruṯu, and shares an activity for teachers and students to explore the connections and patterns in family trees.
URL: teachermagazine.com.au/article