A round pizza is to be divided by three straight cuts, each running right across it. The cuts need not pass through the centre and no piece may be moved between cuts. What is the greatest number of pieces that can be obtained?

Count what each new cut can add. The first cut makes 2 pieces. The second cut can cross the first once, so it passes through 2 pieces and adds 2, making 4. The third cut can cross both earlier cuts, so it passes through 3 pieces and adds 3, making 7. The trap is 6: that is what you get from three cuts all through the centre, the arrangement everybody pictures first, but shifting the cuts off centre so that every pair crosses inside the pizza gains one extra piece. The trap 8 comes from assuming each cut simply doubles the count.