Science Puzzle
How Did Two Teams Dig a Tunnel and Meet in the Middle?
In the 6th century BC, on the Greek island of Samos, an engineer called Eupalinos was given the job of bringing water to the city through a mountain. Two teams started digging at the same time, one on each side, more than a kilometre apart, with hammers, chisels and oil lamps. Deep under the mountain, they met, and the two halves of the tunnel were out by well under a metre.
They didn't have compasses or maps, and there was no way of seeing through rock. How could they be sure of meeting?
The Answer
Geometry. Eupalinos probably measured his way round the mountain in a series of straight, right-angled steps, then used the totals to work out the exact direction of a straight line between the two entrances. Each team then dug along that direction.
Eupalinos didn't leave instructions, so historians have pieced it together from the tunnel itself and from a method described a few centuries later by Hero of Alexandria. Walk round the mountain from one entrance to the other, turning only at right angles, and add up all the steps going north and south, and separately all the steps going east and west. Those two totals are the sides of a right-angled triangle, and the line through the mountain is its long side. Its direction can be marked out with posts at each entrance, and each team can keep checking its tunnel against them.
Height mattered too, or the teams might pass one above the other. Levels could be carried round the mountain with water channels, since still water always lies flat. And Eupalinos seems to have hedged his bets: near the meeting point, one of the tunnels bends off to the side and the passage gets taller, which made it much harder to miss. The final error has been measured at somewhere between 12 and 60 centimetres.
The principle: Surveying with geometry. Measuring a route round an obstacle in right-angled steps gives the direction of the straight line through it, so two teams can dig towards each other without ever seeing each other.