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Converse

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?

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?

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.

Geometric Parabola

Age 14 to 16
Challenge Level
$$y=ax^2+2bx+c$$

Start with $a=1, b=2, c=4$ - what does the graph look like? What key points does it pass through?

Then try $a=2, b=4, c=8$, $a=4, b=8, c=16$ and so on. Look for similarities and differences between the graphs.