Creating and manipulating expressions and formulae

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

If you take four consecutive numbers and add them together, the answer will always be even. What else do you notice?

Make some loops out of regular hexagons. What rules can you discover?

Given a set of points (x,y) with distinct x values, find a polynomial that goes through all of them, then prove some results about the existence and uniqueness of these polynomials.

By proving these particular identities, prove the existence of general cases.

Can you fit polynomials through these points?

Let S1 = 1 , S2 = 2 + 3, S3 = 4 + 5 + 6 ,........ Calculate S17.

If the sides of the triangle in the diagram are 3, 4 and 5, what is the area of the shaded square?