Conjecturing and generalising

Can you find a way of counting the spheres in these arrangements?

15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?

Create a pattern on the small grid. How could you extend your pattern on the larger grid?

Find the relationship between the locations of points of inflection, maxima and minima of functions.

The computer has made a rectangle and will tell you the number of spots it uses in total. Can you find out where the rectangle is?

Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?

Alison and Charlie are playing a game. Charlie wants to go first so Alison lets him. Was that such a good idea?

Can you find some Pythagorean Triples where the two smaller numbers differ by 1?

How can you change the surface area of a cuboid but keep its volume the same? How can you change the volume but keep the surface area the same?