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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?

Label the joints and legs of these graph theory caterpillars so that the vertex sums are all equal.

If a number N is expressed in binary by using only 'ones,' what can you say about its square (in binary)?

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

The Red Express Train usually has five red carriages. How many ways can you find to add two blue carriages?

Bluey-green, white and transparent squares with a few odd bits of shapes around the perimeter. But, how many squares are there of each type in the complete circle? Study the picture and make an estimate.

Try adding together the dates of all the days in one week. Now multiply the first date by 7 and add 21. Can you explain what happens?