# Puzzles

## Archive

Show me a random puzzle**Most recent collections**

#### Sunday Afternoon Maths LXVI

Cryptic crossnumber #2#### Sunday Afternoon Maths LXV

Cryptic crossnumber #1Breaking Chocolate

Square and cube endings

#### Sunday Afternoon Maths LXIV

Equal lengthsDigitless factor

Backwards fours

#### Sunday Afternoon Maths LXIII

Is it equilateral?Cube multiples

List of all puzzles

## Tags

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

The picture below shows two copies of the same rectangle with red and blue lines. The blue line visits the midpoint of the opposite side. The lengths shown in red and blue are of equal length.

What is the ratio of the sides of the rectangle?

## Is it equilateral?

Source: Chalkdust issue 07

In the diagram below, \(ABDC\) is a square. Angles \(ACE\) and \(BDE\) are both 75°.

Is triangle \(ABE\) equilateral? Why/why not?

## Two semicircles

The diagram shows two semicircles.

\(CD\) is a chord of the larger circle and is parallel to \(AB\). The length of \(CD\) is 8m. What is the area of the shaded region (in terms of \(\pi\))?

## 1 December

What is area of the largest area rectangle which will fit in a circle of radius 10?

## Cutting corners

Source: New Scientist Enigma 1773

The diagram below shows a triangle \(ABC\). The line \(CE\) is perpendicular to \(AB\) and the line \(AD\) is perpedicular to \(BC\).

The side \(AC\) is 6.5cm long and the lines \(CE\) and \(AD\) are 5.6cm and 6.0cm respectively.

How long are the other two sides of the triangle?

## Quarter circle

Source: Maths Jam

A quarter circle is drawn in a square. A rectangle is drawn in the corner of the square which touches the circle and has sides of length 8 and 1.

What is the length of a side of the square?

## 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.

## 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.