Visualising and Representing - Short Problems

problem

Favourite

Super shapes

Age

7 to 11

Challenge level

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

problem

Folded a4

Age

11 to 14

Challenge level

What shapes can be made by folding an A4 sheet of paper only once?

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Painted octahedron

Age

11 to 14

Challenge level

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?

problem

Bishop's paradise

Age

11 to 14

Challenge level

Weekly Problem 37 - 2013
Which of the statements about diagonals of polygons is false?

problem

Daniel's star

Age

11 to 14

Challenge level

A solid 'star' shape is created. How many faces does it have?

problem

Squares in a square

Age

11 to 14

Challenge level

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

problem

Potatoes

Age

11 to 14

Challenge level

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?

problem

Soma surface

Age

11 to 14

Challenge level

What is the surface area of the solid shown?

problem

Island hopping

Age

11 to 14

Challenge level

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

problem

Printing error

Age

11 to 14

Challenge level

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

problem

Starting Fibonacci

Age

11 to 14

Challenge level

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

problem

Integral polygons

Age

11 to 14

Challenge level

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?

problem

Fifty coins

Age

11 to 14

Challenge level

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

Age

11 to 14

Challenge level

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

problem

Product and sum

Age

11 to 14

Challenge level

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

problem

Reflected back

Age

11 to 14

Challenge level

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

problem

Blockupied

Age

11 to 14

Challenge level

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

problem

Favourite

Hamiltonian cube

Age

11 to 16

Challenge level

Weekly Problem 36 - 2007
Find the length along the shortest path passing through certain points on the cube.

problem

Doubly symmetric

Age

11 to 14

Challenge level

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?

problem

Night watchmen

Age

11 to 14

Challenge level

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?

problem

Same face

Age

11 to 14

Challenge level

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?

problem

Semaphore

Age

11 to 14

Challenge level

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?

problem

Don't be late

Age

11 to 14

Challenge level

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

problem

Adjacent factors

Age

11 to 14

Challenge level

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

problem

Reading from behind

Age

11 to 14

Challenge level

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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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?

problem

Revolutions

Age

11 to 14

Challenge level

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

problem

Doubly consecutive sums

Age

11 to 14

Challenge level

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

problem

17s and 23s

Age

11 to 14

Challenge level

Can you form this 2010-digit number...

problem

Crawl around the cube

Age

11 to 14

Challenge level

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?

problem

Kangaroo hops

Age

11 to 14

Challenge level

Weekly Problem 11 - 2011
Kanga hops ten times in one of four directions. At how many different points can he end up?

problem

Out of the window

Age

14 to 16

Challenge level

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

problem

Eulerian

Age

14 to 16

Challenge level

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?

problem

Semicircular design

Age

14 to 16

Challenge level

Weekly Problem 9 - 2016
The diagram to the right shows a logo made from semi-circular arcs. What fraction of the logo is shaded?

problem

Rectangle rearrangement

Age

14 to 16

Challenge level

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

problem

Twelve cubed

Age

14 to 16

Challenge level

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?

problem

Dicey directions

Age

14 to 16

Challenge level

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?

problem

Relative time

Age

14 to 16

Challenge level

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

problem

Bike shop

Age

14 to 16

Challenge level

If I walk to the bike shop, but then cycle back, what is my average speed?

problem

Pyramidal n-gon

Age

14 to 16

Challenge level

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?

problem

Cubic covering

Age

14 to 16

Challenge level

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?

problem

Trisected triangle

Age

14 to 16

Challenge level

Weekly Problem 34 - 2015
Four tiles are given. For which of them can three be placed together to form an equilateral triangle?

problem

Travelling by train

Age

14 to 16

Challenge level

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

problem

Oldest and youngest

Age

14 to 16

Challenge level

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?

problem

Travelator

Age

14 to 16

Challenge level

When Andrew arrives at the end of the walkway, how far is he ahead of Bill?

problem

Tennis training

Age

14 to 16

Challenge level

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

problem

Painted purple

Age

14 to 16

Challenge level

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?