# The rabbit in the lake solution

The rabbit in the lake problem

Have a go at the problem. Can the rabbit reach the edge before the man?

 The first idea might be that the rabbit should swim straight for the edge of the lake, on the opposite side to the man. This doesn't work. The rabbit has to swim the radius of the circle (the distance from the middle to the edge). Let's call this r. The man has to travel πr. (If you don't know what π is, click here.) But the man can run four times as fast as the rabbit can swim. That means that he will get there π/4 times as fast as the rabbit. π/4 is less than one (as π is about 3.14159) so the man will get there first, as you can see on the left. Perhaps the rabbit shouldn't swim straight for the edge. Perhaps it could swim in a spiral, so it edges away from the man as well as heading outwards for the edge. This doesn't work either. It works well at first, with the rabbit always keeping the man on the opposite side of the lake. But soon the man starts to catch up the rabbit, which cannot keep him away. In fact, the rabbit starts heading back to the centre of the lake, which doesn't solve anything! So is it impossible? No. We can try combining these two methods. The first method wasn't bad, but the man just beat the rabbit. However, the rabbit could beat the man if it didn't start at the centre, but instead started at a lesser distance from the edge, as long as the man was on the opposite side of the lake. The man would still have πr distance to cover. If the rabbit could travel to the edge faster than this, it would be in with a chance. Well, we've also found out that if the rabbit swims in a spiral, it starts by keeping the man on the opposite side of the lake. In fact, anywhere within a circle of r/4, the rabbit can cheerfully swim round keeping the man opposite it, and even making a little distance towards the edge. When it gets to exactly r/4, it should then change direction, and swim straight for the edge, ignoring the man. It has to travel 3r/4, while the man must travel πr. Remembering that the man travels four times as fast, the time for the rabbit is proportional to 3r/4, and for the man, πr/4. Since 3 is less than π, the rabbit gets to the edge first!