The Great Escape

Launch worksheet

Launch solution 

The Movie

The Great Escape is an all time classic film based on the incredible true story of an escape by British prisoners of war from a German Prisoner of War camp (Stalag Luft III) during World War II. The prisoners have employ all sorts of ingenious tricks and calculations to make the escape happen. Here’s one of the tunnel digging clips from the movie:

Math Scenario 1: Getting Rid of Soil from the Tunnels

Tunnel digging produces a lot of soil and earth (or in this case sand). In normal civilian conditions, you just put the sand somewhere else or in the rubbish. In a prisoner of war camp you have to disguise the fact that you’re digging a tunnel, and make sure the guards don’t notice the sand you’re generating, which looks different to the normal dirt on the ground around the camp.

Multiple tunnels were dug in parallel – we’ll focus on the sand from the “Harry” tunnel. Harry went down about 9 metres, then about 100 metres along under and then out of the camp, and then another 9 metres back up to the surface. It had a cross section of about 0.6 x 0.6 metres, with the occasional larger chamber.

We can work out how much sand would come out of the tunnel:

Total sand volume = tunnel length * average tunnel cross sectional area

Total sand volume = (100 + 9 + 9) * 0.6 * 0.6

Total sand volume = 42.48 m^3

The prisoners devised a clever way to discreetly dump the earth. Small pouches of sand were attached inside the prisoners’ trousers – as they walked around the prison grounds, they could let the sand out over their boots when guards weren’t looking.

Let’s say a prisoner could carry about 1 litre bottle’s worth of sand (or 1000 cubic centimetres) inside their trouser leg per trip. How many prisoner trips are required to dump all the sand from the tunnel?

# trips required = total sand volume / volume per trip

# trips required = 42.48 m^3 / 1000 cm^3 

# trips required = 42.48 m^3 / (1000 cm^3  / 1000000 cm^3 / m^3)

# trips required = 42480

That’s a lot of trips! 

But the prisoner had other ways to get rid of the sand. They were allowed to grow their own small gardens and could dump sand in these as well.

Let’s say there were 40 garden bed areas they could dispose sand in. Each garden bed measured 1 by 2 metres could store a layer of sand 10 centimetres thick below the earth. How much extra dumping capacity does that give the prisoners?

Garden bed sand capacity = # garden beds * garden bed area * sand depth

Garden bed sand capacity = 40 * 2 * 1 * 0.1

Garden bed sand capacity = 8 m^3

This would reduce the number of trips the prisoners dumping sand using their trousers would need to make:

# trips required = total sand volume / volume per trip

# trips required = (42.48 – 8) m^3 / (1000 cm^3  / 1000000 cm^3 / m^3)

# trips required = 34480 trips

Math Scenario 2: Evading Guards

Guards at the prison made regular patrols which made escaping even harder. Timing activities and the escape itself around guard patrols required a lot of research and preparation.

Let’s say during the escape the prisoners observed that a guard would pass by their area every 3 minutes. Let’s also say it was a 100 metre sprint to the trees and to minimize detection they only sent one prisoner at a time, only sending the next one when the first had reached the trees.

How many prisoners could escape between each guard passby?

We need to make an assumption about how fast the prisoner could run the 100 metre sprint to the trees. Weighed down with some gear and disguises, 20 seconds might be a reasonable guess (top sprinters run it in just under 10 seconds).

#number of escapes per guard pass = time available / time per prisoner escape

#number of escapes per guard pass = 3 minutes / 20 seconds

#number of escapes per guard pass = 180 seconds / 20 seconds

#number of escapes per guard pass = 9 prisoners

Real Life Example – Tunnel Boring

Modern tunnel boring operations can be massive. A large tunnel boring machine can have a diameter of 15 metres or more, enabling it to drill multi-lane road tunnels in one pass.

The 57 km long Gotthard Base Tunnel opened in 2016 and goes through  the Swiss Alps. If the tunnel boring machine had a diameter of 10 metres, we can work out the approximate volume of earth and rock that was removed:

Volume removed = tunnel length * tunnel cross sectional area

Volume removed = 57000 m * π * r^2

Volume removed = 57000 m * π * (5m)^2

Volume removed = 4476769 m^3

That’s over 4 million cubic metres of rock and dirt! 

Other tunnels around the world are not quite as long but are even bigger, such as the Channel tunnel between France and the UK. It is 50 km long and consists of two main tunnels that created more than 5 million cubic metres of dirt and rock.