List

Visualising and Representing - Short Problems

Super Shapes
problem
Favourite

Super shapes

Age
7 to 11
Challenge level
filled star empty star empty star

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

Folded A4
problem

Folded a4

Age
11 to 14
Challenge level
filled star empty star empty star
What shapes can be made by folding an A4 sheet of paper only once?
Painted Octahedron
problem

Painted octahedron

Age
11 to 14
Challenge level
filled star empty star empty star
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?
Bishop's Paradise
problem

Bishop's paradise

Age
11 to 14
Challenge level
filled star empty star empty star
Weekly Problem 37 - 2013
Which of the statements about diagonals of polygons is false?
Daniel's Star
problem

Daniel's star

Age
11 to 14
Challenge level
filled star empty star empty star
A solid 'star' shape is created. How many faces does it have?
Squares in a Square
problem

Squares in a square

Age
11 to 14
Challenge level
filled star empty star empty star
In the diagram, the small squares are all the same size. What fraction of the large square is shaded?
Potatoes
problem

Potatoes

Age
11 to 14
Challenge level
filled star empty star empty star
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?
Soma Surface
problem

Soma surface

Age
11 to 14
Challenge level
filled star empty star empty star
What is the surface area of the solid shown?
Island Hopping
problem

Island hopping

Age
11 to 14
Challenge level
filled star empty star empty star
What is the smallest number of ferry trips that Neda needs to take to visit all four islands and return to the mainland?
Printing Error
problem

Printing error

Age
11 to 14
Challenge level
filled star empty star empty star
Every third page number in this book has been omitted. Can you work out what number will be on the last page?
Starting Fibonacci
problem

Starting Fibonacci

Age
11 to 14
Challenge level
filled star empty star empty star

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

Integral Polygons
problem

Integral polygons

Age
11 to 14
Challenge level
filled star filled star empty star
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?
Fifty Coins
problem

Fifty coins

Age
11 to 14
Challenge level
filled star filled star empty star
Cheryl finds a bag of coins. Can you work out how many more 5p coins than 2p coins are in the bag?
Turning N Over
problem

Turning N over

Age
11 to 14
Challenge level
filled star filled star empty star

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

Product and Sum
problem

Product and sum

Age
11 to 14
Challenge level
filled star filled star empty star
When Jim rolled some dice, the scores had the same product and sum. How many dice did Jim roll?
Reflected Back
problem

Reflected back

Age
11 to 14
Challenge level
filled star filled star empty star
Imagine reflecting the letter P in all three sides of a triangle in turn. What is the final result?
Blockupied
problem

Blockupied

Age
11 to 14
Challenge level
filled star filled star empty star
A 1x2x3 block is placed on an 8x8 board and rolled several times.... How many squares has it occupied altogether?
Hamiltonian Cube
problem
Favourite

Hamiltonian cube

Age
11 to 16
Challenge level
filled star filled star empty star
Weekly Problem 36 - 2007
Find the length along the shortest path passing through certain points on the cube.
Doubly Symmetric
problem

Doubly symmetric

Age
11 to 14
Challenge level
filled star filled star empty star
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?
Night Watchmen
problem

Night watchmen

Age
11 to 14
Challenge level
filled star filled star empty star
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?
Same Face
problem

Same face

Age
11 to 14
Challenge level
filled star filled star empty star
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?
Semaphore
problem

Semaphore

Age
11 to 14
Challenge level
filled star filled star empty star
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?
Don't Be Late
problem

Don't be late

Age
11 to 14
Challenge level
filled star filled star empty star
Mary is driving to Birmingham Airport. Using her average speed for the entire journey, find how long her journey took.
Adjacent Factors
problem

Adjacent factors

Age
11 to 14
Challenge level
filled star filled star empty star
Two numbers can be placed adjacent if one of them divides the other. Using only $1,...,10$, can you write the longest such list?
Reading from Behind
problem

Reading from behind

Age
11 to 14
Challenge level
filled star filled star empty star
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?
Hexagon Cut Out
problem

Hexagon cut out

Age
11 to 14
Challenge level
filled star filled star filled star
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?
Revolutions
problem

Revolutions

Age
11 to 14
Challenge level
filled star filled star filled star
Jack and Jill run at different speeds in opposite directions around the maypole. How many times do they pass in the first minute?
Doubly Consecutive Sums
problem

Doubly consecutive sums

Age
11 to 14
Challenge level
filled star filled star filled star
How many numbers less than 2017 are both the sum of two consecutive integers and the sum of five consecutive integers?
17s and 23s
problem

17s and 23s

Age
11 to 14
Challenge level
filled star filled star filled star
Can you form this 2010-digit number...
Crawl Around the Cube
problem

Crawl around the cube

Age
11 to 14
Challenge level
filled star filled star filled star
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?
Kangaroo Hops
problem

Kangaroo hops

Age
11 to 14
Challenge level
filled star filled star filled star
Weekly Problem 11 - 2011
Kanga hops ten times in one of four directions. At how many different points can he end up?
Out of the Window
problem

Out of the window

Age
14 to 16
Challenge level
filled star empty star empty star
Find out how many pieces of hardboard of differing sizes can fit through a rectangular window.
Eulerian
problem

Eulerian

Age
14 to 16
Challenge level
filled star empty star empty star
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?
Semicircular Design
problem

Semicircular design

Age
14 to 16
Challenge level
filled star empty star empty star
Weekly Problem 9 - 2016
The diagram to the right shows a logo made from semi-circular arcs. What fraction of the logo is shaded?
Rectangle Rearrangement
problem

Rectangle rearrangement

Age
14 to 16
Challenge level
filled star empty star empty star
A 3x8 rectangle is cut into two pieces... then rearranged to form a right-angled triangle. What is the perimeter of the triangle formed?
Twelve Cubed
problem

Twelve cubed

Age
14 to 16
Challenge level
filled star empty star empty star
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?
Dicey Directions
problem

Dicey directions

Age
14 to 16
Challenge level
filled star empty star empty star
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?
Relative Time
problem

Relative time

Age
14 to 16
Challenge level
filled star filled star empty star
Albert Einstein is experimenting with two unusual clocks. At what time do they next agree?
Bike Shop
problem

Bike shop

Age
14 to 16
Challenge level
filled star filled star empty star
If I walk to the bike shop, but then cycle back, what is my average speed?
Pyramidal n-gon
problem

Pyramidal n-gon

Age
14 to 16
Challenge level
filled star filled star empty star
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?
Cubic Covering
problem

Cubic covering

Age
14 to 16
Challenge level
filled star filled star empty star
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?
Trisected Triangle
problem

Trisected triangle

Age
14 to 16
Challenge level
filled star filled star empty star
Weekly Problem 34 - 2015
Four tiles are given. For which of them can three be placed together to form an equilateral triangle?
Travelling by Train
problem

Travelling by train

Age
14 to 16
Challenge level
filled star filled star empty star
Stephen stops at Darlington on his way to Durham. At what time does he arrive at Durham?
Oldest and Youngest
problem

Oldest and youngest

Age
14 to 16
Challenge level
filled star filled star empty star
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?
Travelator
problem

Travelator

Age
14 to 16
Challenge level
filled star filled star empty star
When Andrew arrives at the end of the walkway, how far is he ahead of Bill?
Tennis Training
problem

Tennis training

Age
14 to 16
Challenge level
filled star filled star empty star
After tennis training, Andy, Roger and Maria collect up the balls. Can you work out how many Andy collects?
Painted Purple
problem

Painted purple

Age
14 to 16
Challenge level
filled star filled star empty star
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?
Facial Sums
problem

Facial sums

Age
14 to 16
Challenge level
filled star filled star empty star
Can you make the numbers around each face of this solid add up to the same total?
Folding in Half
problem

Folding in half

Age
14 to 16
Challenge level
filled star filled star empty star
How does the perimeter change when we fold this isosceles triangle in half?
Tied up
problem

Tied up

Age
14 to 16
Challenge level
filled star filled star empty star
How much of the field can the animals graze?
Packing Boxes
problem

Packing boxes

Age
14 to 16
Challenge level
filled star filled star empty star
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.
Newspaper Sheets
problem

Newspaper sheets

Age
14 to 16
Challenge level
filled star filled star empty star
From only the page numbers on one sheet of newspaper, can you work out how many sheets there are altogether?
In or Out?
problem

In or out?

Age
14 to 16
Challenge level
filled star filled star filled star
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?
Centre Square
problem

Centre square

Age
14 to 16
Challenge level
filled star filled star filled star
What does Pythagoras' Theorem tell you about the radius of these circles?