January 2003, Stage 4&5

Problems

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LOGO Challenge 7 - More Stars and Squares

Stage: 3 and 4 Challenge Level: Challenge Level:1

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

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Excel Technique: Triangular Arrays by Turning Off Zeros

Stage: 3 and 4 Challenge Level: Challenge Level:2 Challenge Level:2

Learn how to use Excel to create triangular arrays.

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Excel Interactive Resource: Number Grid Functions

Stage: 3 and 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Excel Investigation: Pascal Multiples

Stage: 3 and 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Excel Investigation: Number Pyramids

Stage: 3 and 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Tree Graphs

Stage: 4 Challenge Level: Challenge Level:1

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. . . .

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Take a Square

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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.

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Semi-square

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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?

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Square Pizza

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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?

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A Tilted Square

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

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Summit

Stage: 5 Challenge Level: Challenge Level:1

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

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Cocked Hat

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Sketch the graphs for this implicitly defined family of functions.

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Modular Knights

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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