# January 2003, Stage 4&5

## Problems

### LOGO Challenge 7 - More Stars and Squares

##### Age 11 to 16 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

##### Age 11 to 16 Challenge Level:

Learn how to use Excel to create triangular arrays.

### Excel Interactive Resource: Number Grid Functions

##### Age 11 to 16 Challenge Level:

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

### Excel Investigation: Pascal Multiples

##### Age 11 to 16 Challenge Level:

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

### Excel Investigation: Number Pyramids

##### Age 11 to 16 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

##### Age 14 to 16 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

##### Age 14 to 16 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

##### Age 14 to 16 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

##### Age 14 to 16 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

##### Age 16 to 18 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

##### Age 16 to 18 Challenge Level:

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

### Cocked Hat

##### Age 16 to 18 Challenge Level:

Sketch the graphs for this implicitly defined family of functions.

### Modular Knights

##### Age 16 to 18 Challenge Level:

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