In this short problem, can you deduce the likely location of the odd ones out in six sets of random numbers?

Look for the common features in these graphs. Which graphs belong together?

A polite number can be written as the sum of two or more consecutive positive integers. Find the consecutive sums giving the polite numbers 544 and 424. What characterizes impolite numbers?

Which of these triangular jigsaws are impossible to finish?

Can you make matrices which will fix one lucky vector and crush another to zero?

When does a pattern start to exhibit structure? Can you crack the code used by the computer?