FIRST HALF

**Final match** of **Euro 2012** in **Kiev / Kyiv**. The national teams from **Spain** and **Italy** compete for the championship trophy.

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The referee whistles the final of the match, the winner celebrates the victory, and people start to **leave the stadium**. Because of an electronic problem, the **exit doors** are **blocked**, so that we have only **2 open emergency gates** for the public to go out.

After the initial confusion, people begin to realize the problem and start to leave the stadium. Some people left the stadium in the first minute, and **every minute** there's **outside** the stadium **twice as many people** as in the previous minute.

Finally, after **10 minutes**, the stadium is completely empty.

Imagine that, instead of 2 doors, security guards has managed to open **4 gates**, and people have been leaving the stadium in the **same manner as described**,

**How many minutes** would have been necessary to empty the stadium in this case?

First, it seems that if people have evacuated the stadium in **10 minutes** through **2 gates**, with **4 open doors** the stadium should be empty in **5 minutes** (half the time). But, obviously, that's a wrong calculation.

Let`s go to **minute 10**. Now all the public is out of the stadium. If we said that every minute that passes, people outside the stadium doubles, this means that at the end of **minute 9**, outside the stadium there was **half the people** than at the end of minute 10. Ie half the total spectators of the match.

In other words, at the end of **minute 9**, **half** the public was **outside** the stadium, and the other half was still **inside**. Therefore, if in **9 minutes** half the public has come out through the **2 open doors**, the other half of the public could have come out through **another 2 open doors** in these **9 minutes **. Which means that it takes ** 9 minutes**, and not 5 minutes, as it might seem initially, to leave the stadium if we open 4 gates.

Let's see another way:

Suppose that in the **first minute** **50 people** go out per **gate**, that is, a total of (50 by 2 doors=)** 100** people. After the **second minute **, we double the people outside the stadium (**200**). We continue with the same progression, we'll have **400** people in **3 minutes**, **800** people after **4 minutes**... and so on until to the **minute 10**. Let's see how this is reflected in the following table:

Now suppose that we open **4** **gates**, and in the first minute **50 people** go out per gate (50 by 4 doors = 200 people), as in the previous case. Let's see what happens:

Until **minute 9** we don't have all the people outside the stadium, and not in minute 5 as we initially supposed.

Can you imagine now what will happen if we open **8, 16, 32 or 64 gates**? Curiously, every time we double the open **gates**, we only advance **one minute**!

The key to this puzzle is that our brains display **arithmetic progressions** (linear functions) with no problems, but have many mental difficulties to imagine **geometric sequences**.

That's why we tend to assimilate **geometric progressions** to the **arithmetic progressions**, which leads us to errors such as we had in this example, or as that which led that ancient Indian king to offer a prize to the inventor of chess, in the form of grains of wheat/rice that grew from square to square in geometric progression. But that's another story...

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