# January 2003, Stage 3&4

## Problems

### 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?