mscroggs.co.uk
mscroggs.co.uk

subscribe

Puzzles

Balanced sets

A set of points in the plane is called 'balanced' if for any two points \(A\) and \(B\) in the set, there is another point \(C\) in the set such that \(AC=BC\) (here \(AC\) is the distance between \(A\) and \(C\)).
For all \(n\geq3\), find a balanced set of \(n\) points.

Show answer

Squared circle

Each side of a square has a circle drawn on it as diameter. The square is also inscribed in a fifth circle as shown.
Find the ratio of the total area of the shaded crescents to the area of the square.

Show answer

Two triangles

Source: Maths Jam
The three sides of this triangle have been split into three equal parts and three lines have been added.
What is the area of the smaller blue triangle as a fraction of the area of the original large triangle?

Show answer & extension

Dodexagon

In the diagram, B, A, C, D, E, F, G, H, I, J, K and L are the vertices of a regular dodecagon and B, A, M, N, O and P are the vertices of a regular hexagon.
Show that A, M and E lie on a straight line.

Show answer & extension

Equal side and angle

In the diagram shown, the lengths \(AD = CD\) and the angles \(ABD=CBD\).
Prove that the lengths \(AB=BC\).

Show answer

Arctan

Prove that \(\arctan(1)+\arctan(2)+\arctan(3)=\pi\).

Show answer & extension

Square deal

This unit square is divided into four regions by a diagonal and a line that connects a vertex to the midpoint of an opposite side. What are the areas of the four regions?

Show answer & extension

Half an equilateral triangle

Source: Maths Jam
What is the shortest straight line which bisects an equilateral triangle?

Show answer & extension

Archive

Show me a random puzzle
 Most recent collections 

Advent calendar 2019

Sunday Afternoon Maths LXVII

Coloured weights
Not Roman numerals

Advent calendar 2018

Sunday Afternoon Maths LXVI

Cryptic crossnumber #2

List of all puzzles

Tags

indices polygons trigonometry floors percentages elections area crossnumber routes perimeter differentiation arrows angles volume cryptic clues 3d shapes pascal's triangle mean shapes products sum to infinity geometry range books digital clocks lines grids people maths proportion squares taxicab geometry planes complex numbers factorials time numbers clocks cube numbers dice rectangles multiples cryptic crossnumbers tiling square numbers logic balancing advent integers functions gerrymandering star numbers crossnumbers triangle numbers probability coordinates dodecagons colouring multiplication probabilty christmas addition number sums chocolate hexagons crosswords chess coins bases 2d shapes triangles calculus square roots remainders quadratics ave perfect numbers spheres prime numbers sport ellipses regular shapes graphs games irreducible numbers circles fractions cards dates folding tube maps doubling shape symmetry wordplay surds division rugby factors integration median dominos chalkdust crossnumber speed scales partitions palindromes digits odd numbers unit fractions means averages menace money sequences parabolas algebra the only crossnumber

Archive

Show me a random puzzle
▼ show ▼
© Matthew Scroggs 2012–2020