There are two adults and two children wishing to cross a river. They have a small boat that can only carry one adult, or up to two children at a time. How do all of them get across the river in the fewest steps possible? (The boat can’t be sent back without someone in it.) Show answer
It often helps to draw pictures, or note down what we’re doing in some way so we can keep track of where everyone is, so I’m going to draw a little diagram of the river, with the starting bank on the left, and the destination on the right. I’ll write the letter a for an adult and c for a child to show where they are, and when I want to send someone across the river I’ll circle them and draw an arrow. You might have decided to draw things differently, and that’s ok. If we want to make sure we’re doing this with the fewest steps possible then we need to make sure we’re not backtracking (undoing our last move) at any point, and that there aren’t any possibilities we might have missed which would be quicker.
If a single person goes across first, then the only way to get the boat back would be for them to return, so we’d be back where we started. Therefore, the two children have to go across first, so one of them can take the boat back. Then an adult can go across. (If instead the child went then that would undo the last move.) A child can bring the boat back. Now you might think of taking an adult across, but then an adult would have to bring the boat back, so it would be pointless. The two children have to go across, so that one can bring the boat back. Now you might be thinking: we’ve been here before, with an adult and a child on each bank of the river, so we must have backtracked. But back then the boat was on the right, now it’s on the left, so we have actually advanced. We can take an adult across. And a child can bring the boat back. Now both children can cross the river! This took nine steps.