Working systematically

In this follow-up to the problem Odds and Evens, we invite you to analyse a probability situation in order to find the general solution for a fair game.

Four rods are hinged at their ends to form a convex quadrilateral. Investigate the different shapes that the quadrilateral can take. Be patient this problem may be slow to load.

Can you create a Latin Square from multiples of a six digit number?

Which has the greatest area, a circle or a square, inscribed in an isosceles right angle triangle?

Is it possible to have a tetrahedron whose six edges have lengths 10, 20, 30, 40, 50 and 60 units?

Using a ruler, pencil and compasses only, it is possible to construct a square inside any triangle so that all four vertices touch the sides of the triangle.

A plastic funnel is used to pour liquids through narrow apertures. What shape funnel would use the least amount of plastic to manufacture for any specific volume ?

Use the differences to find the solution to this Sudoku.

Five equations and five unknowns. Is there an easy way to find the unknown values?

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.