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Clearly if a, b and c are the lengths of the sides of an equilateral triangle then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true?

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Consecutive Squares

The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?

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Parabolic Patterns

The illustration shows the graphs of fifteen functions. Two of them have equations y=x^2 and y=-(x-4)^2. Find the equations of all the other graphs.

Quadratic Transformations

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

For the first number plumber, try dropping the same number into both functions. What can you say about the two points that are plotted?
Can you see a relationship between the two number plumbers that explains the relationship between the points?