This is part of our collection of Short Problems.

You may also be interested in our longer problems on Visualising and Representing.

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Favourite

### Super Shapes

The value of the circle changes in each of the following problems. Can you discover its value in each problem?

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### Bishop's Paradise

Weekly Problem 37 - 2013

Which of the statements about diagonals of polygons is false?

Which of the statements about diagonals of polygons is false?

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### Squares in a Square

In the diagram, the small squares are all the same size. What fraction of the large square is shaded?

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### Potatoes

Weekly Problem 19 - 2009

When I looked at the greengrocer's window I saw a sign. When I went in and looked from the other side, what did I see?

When I looked at the greengrocer's window I saw a sign. When I went in and looked from the other side, what did I see?

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### Island Hopping

What is the smallest number of ferry trips that Neda needs to take to visit all four islands and return to the mainland?

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### Printing Error

Every third page number in this book has been omitted. Can you work out what number will be on the last page?

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### Starting Fibonacci

What is the first term of a Fibonacci sequence whose second term is 4 and fifth term is 22?

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### Painted Octahedron

What is the smallest number of colours needed to paint the faces of a regular octahedron so that no adjacent faces are the same colour?

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### Product and Sum

When Jim rolled some dice, the scores had the same product and sum. How many dice did Jim roll?

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### Reflected Back

Imagine reflecting the letter P in all three sides of a triangle in turn. What is the final result?

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### Blockupied

A 1x2x3 block is placed on an 8x8 board and rolled several times.... How many squares has it occupied altogether?

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Favourite

### Hamiltonian Cube

Weekly Problem 36 - 2007

Find the length along the shortest path passing through certain points on the cube.

Find the length along the shortest path passing through certain points on the cube.

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### Doubly Symmetric

What is the smallest number of additional squares that must be shaded so that this figure has at least one line of symmetry and rotational symmetry of order 2?

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### Night Watchmen

Grannie's watch gains 30 minutes every hour, whilst Grandpa's watch loses 30 minutes every hour. What is the correct time when their watches next agree?

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### Same Face

A cube is rolled on a plane, landing on the squares in the order shown. Which two positions had the same face of the cube touching the surface?

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### Semaphore

I am standing behind five pupils who are signalling a five-digit number to someone on the opposite side of the playground. What number is actually being signalled?

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### Don't Be Late

Mary is driving to Birmingham Airport. Using her average speed for the entire journey, find how long her journey took.

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### Adjacent Factors

Two numbers can be placed adjacent if one of them divides the other. Using only $1,...,10$, can you write the longest such list?

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### Reading from Behind

Can you find the time between 3 o'clock and 10 o'clock when my digital clock looks the same from both the front and back?

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### Integral Polygons

Each interior angle of a particular polygon is an obtuse angle which is a whole number of degrees. What is the greatest number of sides the polygon could have?

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### Fifty Coins

Cheryl finds a bag of coins. Can you work out how many more 5p coins than 2p coins are in the bag?

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### Turning N Over

A card with the letter N on it is rotated through two different axes. What does the card look like at the end?

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### Revolutions

Jack and Jill run at different speeds in opposite directions around the maypole. How many times do they pass in the first minute?

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### Doubly Consecutive Sums

How many numbers less than 2017 are both the sum of two consecutive integers and the sum of five consecutive integers?

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### Crawl Around the Cube

Weekly Problem 37 - 2010

An ant is crawling around the edges of a cube. From the description of his path, can you predict when he will return to his starting point?

An ant is crawling around the edges of a cube. From the description of his path, can you predict when he will return to his starting point?

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### Kangaroo Hops

Weekly Problem 11 - 2011

Kanga hops ten times in one of four directions. At how many different points can he end up?

Kanga hops ten times in one of four directions. At how many different points can he end up?

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### Hexagon Cut Out

Weekly Problem 52 - 2012

An irregular hexagon can be made by cutting the corners off an equilateral triangle. How can an identical hexagon be made by cutting the corners off a different equilateral triangle?

An irregular hexagon can be made by cutting the corners off an equilateral triangle. How can an identical hexagon be made by cutting the corners off a different equilateral triangle?

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### Semicircular Design

Weekly Problem 9 - 2016

The diagram to the right shows a logo made from semi-circular arcs. What fraction of the logo is shaded?

The diagram to the right shows a logo made from semi-circular arcs. What fraction of the logo is shaded?

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### Rectangle Rearrangement

A 3x8 rectangle is cut into two pieces... then rearranged to form a right-angled triangle. What is the perimeter of the triangle formed?

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### Twelve Cubed

A wooden cube with edges of length 12cm is cut into cubes with edges of length 1cm. What is the total length of the all the edges of these centimetre cubes?

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### Dicey Directions

An ordinary die is placed on a horizontal table with the '1' face facing East... In which direction is the '1' face facing after this sequence of moves?

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### Out of the Window

Find out how many pieces of hardboard of differing sizes can fit through a rectangular window.

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### Eulerian

Weekly Problem 37 - 2014

Which of the five diagrams below could be drawn without taking the pen off the page and without drawing along a line already drawn?

Which of the five diagrams below could be drawn without taking the pen off the page and without drawing along a line already drawn?

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### Cubic Covering

A blue cube has blue cubes glued on all of its faces. Yellow cubes are then glued onto all the visible blue facces. How many yellow cubes are needed?

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### Trisected Triangle

Weekly Problem 34 - 2015

Four tiles are given. For which of them can three be placed together to form an equilateral triangle?

Four tiles are given. For which of them can three be placed together to form an equilateral triangle?

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### Travelling by Train

Stephen stops at Darlington on his way to Durham. At what time does he arrive at Durham?

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### Oldest and Youngest

Edith had 9 children at 15 month intervals. If the oldest is now six times as old as the youngest, how old is her youngest child?

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### Tennis Training

After tennis training, Andy, Roger and Maria collect up the balls. Can you work out how many Andy collects?

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### Painted Purple

Three faces of a $3 \times 3$ cube are painted red, and the other three are painted blue. How many of the 27 smaller cubes have at least one red and at least one blue face?

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### Facial Sums

Can you make the numbers around each face of this solid add up to the same total?

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### Packing Boxes

Look at the times that Harry, Christine and Betty take to pack boxes when working in pairs, to find how fast Christine can pack boxes by herself.

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### Newspaper Sheets

From only the page numbers on one sheet of newspaper, can you work out how many sheets there are altogether?

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### Relative Time

Albert Einstein is experimenting with two unusual clocks. At what time do they next agree?

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### Pyramidal n-gon

The base of a pyramid has n edges. What is the difference between the number of edges the pyramid has and the number of faces the pyramid has?

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### In or Out?

Weekly Problem 52 - 2014

Four arcs are drawn in a circle to create a shaded area. What fraction of the area of the circle is shaded?

Four arcs are drawn in a circle to create a shaded area. What fraction of the area of the circle is shaded?