Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?
I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?
How many noughts are at the end of these giant numbers?
How many different solutions can you find to this problem?
Arrange 25 officers, each having one of five different ranks
$a$, $b$, $c$, $d$ and $e$, and belonging to one of five different
regiments $p$, $q$, $r$, $s$ and $t$, in a square formation 5 by 5,
so that each row and each file contains just one officer of each
rank and just one from each regiment.