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### Number and algebra

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# Visualising and Representing - Short Problems

### Super Shapes

### Daniel's Star

### Island Hopping

### Starting Fibonacci

### Squares in a Square

### Bishop's Paradise

### Folded A4

### Painted Octahedron

### Printing Error

### Potatoes

### Integral Polygons

### Fifty Coins

### Semaphore

### Turning N Over

### Adjacent Factors

### Product and Sum

### Reflected Back

### Night Watchmen

### Don't Be Late

### Same Face

### Blockupied

### Reading from Behind

### Doubly Symmetric

### Doubly Consecutive Sums

### Kangaroo Hops

### Revolutions

### Crawl Around the Cube

### Hexagon Cut Out

### Hamiltonian Cube

### Rectangle Rearrangement

### Twelve Cubed

### Out of the Window

### Eulerian

### Dicey Directions

### Semicircular Design

### Packing Boxes

### Travelling by Train

### Newspaper Sheets

### Painted Purple

### Oldest and Youngest

### Relative Time

### Travelator

### Folding in Half

### Cubic Covering

### Trisected Triangle

### Bike Shop

### Pyramidal N-gon

### Tennis Training

### Facial Sums

### Centre Square

### In or Out?

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This is part of our collection of Short Problems.

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

Age 7 to 11

ShortChallenge Level

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

Age 11 to 14

ShortChallenge Level

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

Age 11 to 14

ShortChallenge Level

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

Age 11 to 14

ShortChallenge Level

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

Age 11 to 14

ShortChallenge Level

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

Age 11 to 14

ShortChallenge Level

Weekly Problem 37 - 2013

Which of the statements about diagonals of polygons is false?

Age 11 to 14

ShortChallenge Level

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

Age 11 to 14

ShortChallenge 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?

Age 11 to 14

ShortChallenge Level

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

Age 11 to 14

ShortChallenge 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?

Age 11 to 14

ShortChallenge 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?

Age 11 to 14

ShortChallenge Level

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

Age 11 to 14

ShortChallenge 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?

Age 11 to 14

ShortChallenge Level

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

Age 11 to 14

ShortChallenge 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?

Age 11 to 14

ShortChallenge Level

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

Age 11 to 14

ShortChallenge Level

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

Age 11 to 14

ShortChallenge 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?

Age 11 to 14

ShortChallenge Level

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

Age 11 to 14

ShortChallenge 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?

Age 11 to 14

ShortChallenge Level

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

Age 11 to 14

ShortChallenge 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?

Age 11 to 14

ShortChallenge 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?

Age 11 to 14

ShortChallenge Level

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

Age 11 to 14

ShortChallenge Level

Weekly Problem 11 - 2011

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

Age 11 to 14

ShortChallenge Level

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

Age 11 to 14

ShortChallenge 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?

Age 11 to 14

ShortChallenge Level

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?

Age 11 to 16

ShortChallenge Level

Weekly Problem 36 - 2007

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

Age 14 to 16

ShortChallenge 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?

Age 14 to 16

ShortChallenge 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?

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge 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?

Age 14 to 16

ShortChallenge 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?

Age 14 to 16

ShortChallenge 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?

Age 14 to 16

ShortChallenge Level

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.

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge 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?

Age 14 to 16

ShortChallenge 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?

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

How does the perimeter change when we fold this isosceles triangle in half?

Age 14 to 16

ShortChallenge 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?

Age 14 to 16

ShortChallenge Level

Weekly Problem 34 - 2015

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

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge 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?

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

What does Pythagoras' Theorem tell you about the radius of these circles?

Age 14 to 16

ShortChallenge Level

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?