Reasoning, convincing and proving

In a league of 5 football teams which play in a round robin tournament show that it is possible for all five teams to be league leaders.

Blue Flibbins are so jealous of their red partners that they will not leave them on their own with any other bue Flibbin. What is the quickest way of getting the five pairs of Flibbins safely to the new planet?

In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...

In how many ways can you arrange three dice side by side on a surface so that the sum of the numbers on each of the four faces (top, bottom, front and back) is equal?

Baker, Cooper, Jones and Smith are four people whose occupations are teacher, welder, mechanic and programmer, but not necessarily in that order. What is each person’s occupation?

Your partner chooses two beads and places them side by side behind a screen. What is the minimum number of guesses you would need to be sure of guessing the two beads and their positions?

How many winning lines can you make in a three-dimensional version of noughts and crosses?

Can you produce convincing arguments that a selection of statements about numbers are true?

An equilateral triangle is constructed on BC. A line QD is drawn, where Q is the midpoint of AC. Prove that AB // QD.

Investigate the number of points with integer coordinates on circles with centres at the origin for which the square of the radius is a power of 5.