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Converse

Clearly if a, b and c are the lengths of the sides of a triangle and the triangle is equilateral then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true, and if so can you prove it? That is if a^2 + b^2 + c^2 = ab + bc + ca is the triangle with side lengths a, b and c necessarily equilateral?

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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.

Exploring Quadratic Mappings

Stage: 4 Challenge Level: Challenge Level:1

Preveina from Crest Girls' Academy made the following observation:

When I put the input as 0, I perceived that that the output is the product of the numbers that I have replaced. For example, for the initial values of 1 and 3, when the input was 0 the output was 3. When I changed the values in the mapping to 8 and 5, the input of 0 gave an output of 40.

Rajeev from Fair Field Junior School sent us his explorations with quadratic mappings, which you can read here.