Visualising - Short Problems
Folded A4
What shapes can be made by folding an A4 sheet of paper only once?
Squares in a Square
In the diagram, the small squares are all the same size. What fraction of the large square is shaded?
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?
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?
Daniel's Star
A solid 'star' shape is created. How many faces does it have?
Printing Error
Every third page number in this book has been omitted. Can you work out what number will be on the last page?
Bishop's Paradise
Weekly Problem 37 - 2013
Which of the statements about diagonals of polygons is false?
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?
Blockupied
A 1x2x3 block is placed on an 8x8 board and rolled several times.... How many squares has it occupied altogether?
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?
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?
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?
Reflected Back
Imagine reflecting the letter P in all three sides of a triangle in turn. What is the final result?
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?
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?
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?
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?
Kangaroo Hops
Weekly Problem 11 - 2011
Kanga hops ten times in one of four directions. At how many different points can he end up?
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?
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?
Hamiltonian Cube
Weekly Problem 36 - 2007
Find the length along the shortest path passing through certain points on the cube.
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?
Out of the Window
Find out how many pieces of hardboard of differing sizes can fit through a rectangular window.
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?
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?
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?
Newspaper Sheets
From only the page numbers on one sheet of newspaper, can you work out how many sheets there are altogether?
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?
Folding in Half
How does the perimeter change when we fold this isosceles triangle in half?
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?
Trisected Triangle
Weekly Problem 34 - 2015
Four tiles are given. For which of them can three be placed together to form an equilateral triangle?
Facial Sums
Can you make the numbers around each face of this solid add up to the same total?
Centre Square
What does Pythagoras' Theorem tell you about the radius of these circles?
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?
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Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?