Conjecturing and generalising

What is the largest number of circles we can fit into the frame without them overlapping? How do you know? What will happen if you try the other shapes?

ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.

Put the numbers 1, 2, 3, 4, 5, 6 into the squares so that the numbers on each circle add up to the same amount. Can you find the rule for giving another set of six numbers?

A polite number can be written as the sum of two or more consecutive positive integers, for example 8+9+10=27 is a polite number. Can you find some more polite, and impolite, numbers?

Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?

While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book?

Consider all two digit numbers (10, 11, . . . ,99). In writing down all these numbers, which digits occur least often, and which occur most often ? What about three digit numbers, four digit numbers and so on?

Your data is a set of positive numbers. What is the maximum value that the standard deviation can take?

Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?

Can you explain how this card trick works?