# January 2003, All Stages

## Problems

### Little Squares

##### Stage: 1 Challenge Level:

Look at the squares in this problem. What does the next square look like? I draw a square with 81 little squares inside it. How long and how wide is my square?

### What's Left?

##### Stage: 1 Challenge Level:

Use this grid to shade the numbers in the way described. Which numbers do you have left? Do you know what they are called?

### Tangrams

##### Stage: 1 Challenge Level:

Can you make five differently sized squares from the tangram pieces?

### Polydron

##### Stage: 2 Challenge Level:

This activity investigates how you might make squares and pentominoes from Polydron.

### Counter Ideas

##### Stage: 2 Challenge Level:

Here are some ideas to try in the classroom for using counters to investigate number patterns.

### Cycling Squares

##### Stage: 2 Challenge Level:

Can you make a cycle of pairs that add to make a square number using all the numbers in the box below, once and once only?

### Two Primes Make One Square

##### Stage: 2 Challenge Level:

Can you make square numbers by adding two prime numbers together?

### Geoboards

##### Stage: 2 Challenge Level:

This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.

### Penta Primes

##### Stage: 2 Challenge Level:

Using all ten cards from 0 to 9, rearrange them to make five prime numbers. Can you find any other ways of doing it?

### Seven Square Numbers

##### Stage: 2 Challenge Level:

Add the sum of the squares of four numbers between 10 and 20 to the sum of the squares of three numbers less than 6 to make the square of another, larger, number.

### Overlap

##### Stage: 3 Challenge Level:

A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .

### Squaring the Circle

##### Stage: 3 Challenge Level:

Bluey-green, white and transparent squares with a few odd bits of shapes around the perimeter. But, how many squares are there of each type in the complete circle? Study the picture and make. . . .

### LOGO Challenge 7 - More Stars and Squares

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

Can you use LOGO to create a systematic reproduction of a basic design? An introduction to variables in a familiar setting.

### Excel Technique: Triangular Arrays by Turning Off Zeros

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

Learn how to use Excel to create triangular arrays.

### Excel Interactive Resource: Number Grid Functions

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

Use Excel to investigate the effect of translations around a number grid.

### Excel Investigation: Pascal Multiples

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

This spreadsheet highlights multiples of numbers up to 20 in Pascal's triangle. What patterns can you see?

### Excel Investigation: Number Pyramids

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

Use Excel to create some number pyramids. How are the numbers in the base line related to each other? Investigate using the spreadsheet.

### Take a Square

##### Stage: 4 Challenge Level:

Cut off three right angled isosceles triangles to produce a pentagon. With two lines, cut the pentagon into three parts which can be rearranged into another square.

### Semi-square

##### Stage: 4 Challenge Level:

What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?

### Square Pizza

##### Stage: 4 Challenge Level:

Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?

### A Tilted Square

##### Stage: 4 Challenge Level:

The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?

### Tree Graphs

##### Stage: 5 Challenge Level:

A connected graph is a graph in which we can get from any vertex to any other by travelling along the edges. A tree is a connected graph with no closed circuits (or loops. Prove that every tree has. . . .

### Summit

##### Stage: 5 Challenge Level:

Prove that the sum from t=0 to m of (-1)^t/t!(m-t)! is zero.

### Cocked Hat

##### Stage: 5 Challenge Level:

Sketch the graphs for this implicitly defined family of functions.

### Modular Knights

##### Stage: 5 Challenge Level:

Try to move the knight to visit each square once and return to the starting point on this unusual chessboard.