# Puzzles

## Archive

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

#### Sunday Afternoon Maths LXVII

Coloured weightsNot Roman numerals

#### Advent calendar 2018

#### Sunday Afternoon Maths LXVI

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

Cryptic crossnumber #1Breaking Chocolate

Square and cube endings

List of all puzzles

## Tags

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

Today's number is the area of the largest dodecagon that it's possible to fit inside a circle with area \(\displaystyle\frac{172\pi}3\).

## 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?

## 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.

## Dartboard

Concentric circles with radii 1, \(\frac{1}{2}\), \(\frac{1}{3}\), \(\frac{1}{4}\), ... are drawn. Alternate donut-shaped regions are shaded.

What is the total shaded area?

## Circles

Which is largest, the red or the blue area?

#### Show answer & extension

Let \(4x\) be the side length of the square. This means that the radius of the red circle is \(2x\) and the radius of a blue circle is \(x\). Therefore the area of the red circle is \(4\pi x^2\).

The area of one of the blue squares is \(\pi x^2\) so the blue area is \(4\pi x^2\). Therefore

**the two areas are the same**.#### Extension

Is the red or blue area larger?

## Semi circle in a triangle

Source: UKMT Pink Kangaroo 2012

This right-angled triangle above has sides of lengths 12cm, 5cm and 13cm. The diameter of the semicircle lies on the 12cm side and the 13cm side is a tangent to the circle. What is the radius of the semi circle?