 Solutions for Mathematics Enrichment

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# Mathematical Papers

“But the pleasure of learning and knowing, though not the keenest, is yet the least perishable of pleasures; the least subject to external things, and the play of chance, and the wear of time. And as a prudent man puts money by to serve as provision for the material wants of his old age, so too he needs to lay up against the end of his days provision for the intellect.”—A E Houseman

## Published articles

### Proofs by folding of the angle sum of a triangle

The Mathematical Gazette, Volume 102 (March 2018)

• Can the well-known result that ‘the (interior) angles of a triangle add up to 180°’ be proved by folding the triangle?
• This article gives two such proofs.

### Four congruent right triangles

The Mathematical Gazette, Volume 102 (March 2018)

• When four congruent right triangles are arranged so that the hypotenuses form a square, that four points are collinear is not well-known.
• What happens when the four hypotenuses form a rhombus? This article gives a proof of this generalisation.

### The Cairo tiling

The Scottish Mathematicsl Council Journal 47 (December 2017)

• Discusses various properties of the Cairo tiling (the dual of an Archimidean tiling).

### Generalising an SMC problem

The Mathematical Gazette, Volume 101 (November 2016)

• Question 21 from the Senior Mathematical Challenge of November 2015 was a problem about a point $$M$$ in a triangle with sides of length 2, 3 and 4. (The SMC is a competition organised by the United Kingdom Mathematics Trust)
• This article generalises to any side lengths, and considers what happens when the point $$M$$ is replaced by a triangle.

### Generalising the arbelos

The Mathematical Gazette, Volume 100 (July 2016)

• Archimedes was the first to describe the arbelos---a region enclosed by three semicircular arcs---a shape which has been extensively studied.
• This article generalises to a region enclosed by three circular arcs which are not necessarily semicircles.

### Where is the centre of a polygon?

The Mathematical Gazette, Volume 99 (March 2015)

• It is clear which point to take as the centre of a rectangle, say. But where is the “centre” of a general polygon?
• This article discusses the difficulties, and describes one attempt to provide an answer.

### Knots in the classroom

Mathematics in School (March 2015)

• Written jointly with Meike Akveld.
• Describes some elementary topics in knot theory, and explains how these may be used in the classroom.

### Shapes with a skew square as boundary

The Mathematical Gazette, Volume 97 (July 2013)

• What is the most symmetric quadrilateral in $$\mathbb R^3$$? One answer is the skew square $$\mathcal{S}$$, which joins four non-adjacent vertices of a cube.
• This article demonstrates four examples of two-dimensional regions that not only have $$\mathcal{S}$$ as boundary, but which also have the same symmetry as $$\mathcal{S}$$.

### Folding a triangle

The Mathematical Gazette, Volume 97 (July 2013)

• Fold the vertex $$C$$ of triangle $$ABC$$ onto side $$AB$$. Where should one fold in order to minimise the area of the folded corner?
• A geometrical solution is given for the case when $$C$$ is a right angle, and the more general case is analysed.

### Dissecting pentominoes

Mathematics in School (March 2013)

• Can one shape be dissected into another?
• If so, then what is the smallest number of pieces needed?
• This article tries to answer these questions in the case when the shapes are a pentomino and a square. Along the way a connection with Pythagoras’ theorem is given.
• The article also discusses whether the pieces can be hinged together.

### Two semicircles fill half a circle

The Mathematical Gazette, Volume 95 (November 2011)

• Proves the surprising result that two touching semicircles, whose diameters are parallel chords of a circle, occupy half the area of the circle whatever the lengths of the chords are.

### The Dance of the Foci Morphs into a Strophoid

The College Mathematics Journal, Volume 42 (September 2011)

• The intersection of a plane and a cone is a conic section and rotating the plane leads to a family of conics. What happens to the foci of these conics as the plane rotates?
• A classical result gives the locus of the foci as an oblique strophoid when the plane rotates about a tangent to the cone; rotation about a different axis gives a very different curve. This article relates the curves by analyzing the family obtained as the axis of rotation moves.

### Proofs by dissection of a dodecagon

The Mathematical Gazette, Volume 95 (March 2011)

• Describes a solution to an Olympiad problem; the solution relates the problem to a result about the area of a dodecagon.
• Gives two dissection proofs of the dodecagon result.

### Folding paper

The Scottish Mathematicsl Council Journal 40 (December 2010)

• The artistic side of paper-folding (origami) is well-known, but it is less well known that there are mathematical aspects too, and many of them. This article draws attention to some of this mathematics, and shows how folding paper may be helpful in teaching.

### Sudoku is four-dimensional

The Mathematical Gazette, Volume 94 (July 2010)

• Shows that the standard 9×9 Sudoku puzzle is naturally a four-dimensional configuration.
• Enumerates the number of grids where six constraints are imposed rather than the usual three.

### Geometry by numbers

The Scottish Mathematicsl Council Journal 39 (December 2009)

• Shows how to use complex numbers to prove geometrical results, including Varignon’s theorem, van Aubel’s theorem, Thébault’s ﬁrst theorem and a remarkable result about the diagonals of a regular polygon.

### A surprising relationship?

The Mathematical Gazette, Volume 91 (March 2007)

• Considers the accuracy of a scatterplot of the area versus perimeter of random rectangles.

### Dissecting a triangle into a rectangle

The Mathematical Gazette, Volume 89 (November 2005)

• Describes a general procedure for dissecting any triangle into a rectangle with one side given.

### Dissections

Maths Challenges News, Issue 9 (May 2001).

• Considers dissection problems and demonstrates how they may be solved by superimposing tessellations.

### Playing with Dice

Maths Challenges News, Issue 8 (February 2001)

• Discusses unusual dice that behave like normal ones.
• Describes a set of non-transitive dice.

### Parity

Maths Challenges News, Issue 5 (September 1999).

• Explains how a parity argument may be used to prove that something is impossible.

### The volume of the $$n$$-ball

The Mathematical Gazette, Volume 82 (March 1998)

• Derives a formula for the volume of the $$n$$-dimensional ball.

The Mathematical Gazette, Volume 81 (July 1997)

• Demonstrates the necessary and sufficient conditions for the vertices of a quadrilateral to lie on the perimeter of a square.

### Fair means

The Mathematical Gazette, Volume 81 (July 1997)

• Describes an approach to combining marks which in a well-defined sense is as fair as possible.

### Chords, tangents and cubics

The Mathematical Gazette, Volume 79 (July 1995)

• Discusses the relationship between the roots of a cubic equation and the intersections of a cubic curve with a chord or a tangent.

### A polyhedron and its volume

The Mathematical Gazette, Volume 68 (October 1984)

• Illustrates the polyhedron obtained when 24 symmetrically placed tetrahedra are removed from a cube.
• Calculates the volume of the polyhedron.

## Miscellany

A collection of resources for teachers, unpublished articles and notes, made freely available.

### Three squares.

• A classic problem concerning angles in an arrangement of three equal squares.

### Enumerating nets.

• How many nets does a given polyhedron have?
• A familiar result is that there are eleven nets of a cube. How do you prove this?

### Rings of regular polygons.

• A paper inspired by Leon van den Broek.
• Some regular polygons can be arranged in a symmetrical ring. This paper answers the questions, how many such rings are there?

### An elementary proof of the converse of Anne’s theorem

• Suppose that you are given a quadrilateral which is not a parallelogram. Select a point inside the quadrilateral and join it to the four vertices, thus dividing the quadrilateral into four triangles. If the point lies on the line joining the midpoints of the diagonals, then the sum of opposite triangle areas is half the quadrilateral area.

### How to fold simple shapes from A4 paper

• Starting with a single sheet of paper, how do you fold simple shapes like an equilateral triangle? Instructions are given for folding:
• a square
• an equilateral triangle
• a rhombus
• a regular hexagon
• a kite
• Explains why the methods work.

### A graph with a unique Hamiltonian circuit

• A path which visits every vertex of a graph just once and returns to the starting point is known as a Hamiltonian circuit. How many Hamiltonian circuits does a given graph have?
• Shows that for a particular graph the Hamiltonian circuit is unique up to symmetry.

### Chords and regions

• How many regions are created when chords are drawn in a circle?
• Shows the dangers of ‘pattern spotting’—deriving a formula from limited data—and gives two proofs of the correct result.

### Trigonometry and the nonagon

• Stimulated by the article Three trigonometric results from a regular nonagon by David Miles and Chris Pritchard in Mathematics in School (November 2008).
• Shows that each of the results discussed in the article is related to a more general result, though in very different ways.

### The central region of the Mandelbrot set

• Shows that the central region of the Mandelbrot set is a cardioid.

### A derivation of the quadratic formula

• A derivation of the quadratic formula from the relationships between the roots and coefficients (Vieta’s formulae).

### BMO 2004/5 Round 1 question 5

• Describes a solution using continued fractions to a British Mathematical Olympiad problem.

### Langley’s triangle problem

• A solution to Langley’s 20°/80°/80° isosceles triangle problem.

### Newton’s derivation of Kepler’s second law

• Explains the elementary derivation that Newton gave of Kepler’s second law of planetary motion.

### A generalisation of Pythagoras’ theorem

• A generalisation of Pythagoras’ theorem to squares of areas in three dimensions.
• The proof only involves the standard two-dimensional theorem.

### Times of flight in a resisting medium

• An elementary proof that for vertical motion in a resisting medium the time of ascent is less than the time of descent, whatever the law of resistance.

## Presentations

Resources for teachers, made freely available.

For projector presentations, we recommend downloading the PDF and viewing in Full Screen mode.

3D images can be manipulated when the PDF is viewed in Adobe Reader (or the Adobe Reader Plug-in for Web Browsers).

### Spherical geometry

• What is the area of a spherical triangle?
• Contains 3D images showing a lune, a spherical triangle, and the surface of a sphere divided into regions. These help to explain how the area of a spherical triangle can be calculated from its angles.
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