# Visualising - Lower Secondary

Visualising is part of our Thinking Mathematically collection.

### Pick's Theorem

##### Stage: 3 Challenge Level:

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

### Attractive Tablecloths

##### Stage: 4 Challenge Level:

Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?

### Painted Cube

##### Stage: 3 Challenge Level:

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

### Square It

##### Stage: 1, 2, 3 and 4 Challenge Level:

Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.

### Route to Infinity

##### Stage: 3 and 4 Challenge Level:

Can you describe this route to infinity? Where will the arrows take you next?

### Diamond Collector

##### Stage: 3 Challenge Level:

Collect as many diamonds as you can by drawing three straight lines.

### Cops and Robbers

##### Stage: 2 and 3 Challenge Level:

Can you find a reliable strategy for choosing coordinates that will locate the robber in the minimum number of guesses?

### Diminishing Returns

##### Stage: 3 Challenge Level:

In this problem, we have created a pattern from smaller and smaller squares. If we carried on the pattern forever, what proportion of the image would be coloured blue?

### Marbles in a Box

##### Stage: 3 and 4 Challenge Level:

In a three-dimensional version of noughts and crosses, how many winning lines can you make?

### Nine Colours

##### Stage: 3 and 4 Challenge Level:

You have 27 small cubes, 3 each of nine colours. Use the small cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of every colour.

### Vector Journeys

##### Stage: 4 Challenge Level:

Charlie likes to go for walks around a square park, while Alison likes to cut across diagonally. Can you find relationships between the vectors they walk along?

### Frogs

##### Stage: 2 and 3 Challenge Level:

How many moves does it take to swap over some red and blue frogs? Do you have a method?

### Rolling Around

##### Stage: 3 Challenge Level:

A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle?

### Folded A4

##### Stage: 3 and 4 Short Challenge Level:

Weekly Problem 45 - 2011
What shapes can be made by folding an A4 sheet of paper only once?

### Regional Division

##### Stage: 3 and 4 Short Challenge Level:

Weekly Problem 52 - 2011
Draw two intersecting rectangles on a sheet of paper. How many regions are enclosed? Can you find the largest number of regions possible?

### Out the Window

##### Stage: 3 and 4 Short Challenge Level:

Weekly Problem 3 - 2012
Find out how many pieces of hardboard of differing sizes can fit through a rectangular window.

##### Stage: 2 and 3 Short Challenge Level:

Weekly Problem 17 - 2012
Which shapes can be made by folding a piece of A4 paper?

### Hexagon Cut Out

##### Stage: 3 Short Challenge 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?

### Turning N Over

##### Stage: 2 and 3 Short Challenge Level:

Weekly Problem 34 - 2013
A card with the letter N on it is rotated through two different axes. What does the card look like at the end?

### Sewing

##### Stage: 3 Short Challenge Level:

Weekly Problem 20 - 2006
A single piece of string is threaded through five holes on a piece of card. How is this possible?

### Blockupied

##### Stage: 3 and 4 Short Challenge Level:

Weekly Problem 28 - 2007
A 1x2x3 block is placed on an 8x8 board and rolled several times.... How many squares has it occupied altogether?

### Sliding Coins

##### Stage: 3 and 4 Short Challenge Level:

Weekly Problem 43 - 2007
The diagram shows 10 identical coins which fit exactly inside a wooden frame. What is the largest number of coins that may be removed so that each remaining coin is still unable to slide.

##### Stage: 3 and 4 Short Challenge Level:

Weekly Problem 46 - 2007
When a solid cube is held up to the light, how many of the shapes shown could its shadow have?

### Dicey Directions

##### Stage: 3 and 4 Short Challenge Level:

Weekly Problem 25 - 2008
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?

### ACE, TWO, THREE...

##### Stage: 3 Challenge Level:

Can you picture how to order the cards to reproduce Charlie's card trick for yourself?

### Cubic Net

##### Stage: 4 and 5 Challenge Level:

This is an interactive net of a Rubik's cube. Twists of the 3D cube become mixes of the squares on the 2D net. Have a play and see how many scrambles you can undo!

### Potatoes

##### Stage: 3 Short 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?

### Semaphore

##### Stage: 3 Short Challenge Level:

Weekly Problem 34 - 2009
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?

### Air Nets

##### Stage: 2, 3, 4 and 5 Challenge Level:

Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.

### Squarely in the Grid

##### Stage: 2 and 3 Short Challenge Level:

Weekly Problem 47 - 2009
How many squares can you draw on this lattice?

### Painted Purple

##### Stage: 3 Short Challenge Level:

Weekly Problem 18 - 2010
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?

### Crawl Around the Cube

##### Stage: 3 Short 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?

##### Stage: 2 and 3 Short Challenge Level:

Weekly Problem 24 - 2011
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?

### Newspaper Sheets

##### Stage: 2 and 3 Short Challenge Level:

Weekly Problem 29 - 2011
From only the page numbers on one sheet of newspaper, can you work out how many sheets there are altogether?

### Constructing Triangles

##### Stage: 3 Challenge Level:

Generate three random numbers to determine the side lengths of a triangle. What triangles can you draw?

### Seven Squares

##### Stage: 3 and 4 Challenge Level:

Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?

### Painted Octahedron

##### Stage: 3 and 4 Short Challenge Level:

Weekly Problem 4 - 2014
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?

### In or Out?

##### Stage: 4 Short Challenge 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?

### Gaudi's Design

##### Stage: 3 and 4 Short Challenge Level:

Weekly Problem 12 - 2015
Eight lines are drawn in a regular octagon to form a pattern. What fraction of the octagon is shaded?

### Reflected Back

##### Stage: 3 and 4 Short Challenge Level:

Weekly Problem 18 - 2015
Beatrix relfects the letter P in all three sides of a triangle in turn. What is the final result?

### Trisected Triangle

##### Stage: 4 Short Challenge Level:

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

### Semicircular Design

##### Stage: 3 and 4 Short 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?

### Doubly Symmetric

##### Stage: 3 and 4 Short Challenge Level:

Weekly Problem 24 - 2016
What is the smallest number of additional lines that must be shaded so that this figure has at least one line of symmetry and rotational symmetry of order 2?

### Squares in a Square

##### Stage: 3 Short Challenge Level:

Weekly Problem 43 - 2016
In the diagram, the small squares are all the same size. What fraction of the large square is shaded?

### Same Face

##### Stage: 3 and 4 Short Challenge Level:

Weekly Problem 27 - 2017
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?

### Soma Surface

##### Stage: 3 and 4 Short Challenge Level:

Weekly Problem 48 - 2017
What is the surface area of the solid shown?