FIRST HALF

**England** and **Brazil** are playing a **friendly football match**. Few minutes before the match starts, players from both teams are **warming up** on the pitch.

vs. |

English players are running **from sideline to sideline**, according to the **instructions** their **trainer** has given them, and which are as follows:

- Every second a player is given the starting signal from the sideline.

- They must complete the distance to the other sideline in 10 seconds, at a steady pace, following the foregoing player.

- Once a player gets to the other sideline, and without stopping, he will turn around and return to the original sideline at the same steady pace as in the forward way (10 seconds), and then he can go to the changing room.

**Wayne Rooney** is the **last** player to leave out. Just at the moment the trainer gives the starting signal, **Peter Crouch** reaches the sideline.

Can you **estimate** how many **players** **Ronney** will meet on his way **to the other sideline**?

Some people answer that **Rooney** crosses **10 players**: if he takes **10 seconds** to get to the other side, and **every second a player has left out** the band, we do the **cocient** and we get this result: **10 players**.

Other people think the same way, but they add, to the **10 players** we have calculated, **one more** player, corresponding to **Peter Crouch**, who crosses at **second 0:00**. So they stand that **Rooney** will cross **11 players**. And another group of people answer that he will meet with a larger or smaller number of players.

Let's see who's right.

The second option (**11 players**) is incorrect, because at the 10th second **Rooney** will not match anyone on the other side, that is, he will meet players at the seconds 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, but not at second 10, since the previous player began his return way one second ago.

So, at first sight, the first reasoning seems correct. We have the **time** (**10 seconds**), the output **frequency** (1 player / second) and we obtain the **result**:

And this would be correct if **Rooney** still **remains** in the band. But here we must consider a **third factor** that we've fogotten: that **Rooney** is **moving** during these 10 seconds.

This means that **Rooney** will not meet a player every second of his route, but as both players are moving toward each other, they will complete the distance between them in **half the time** (0.5 seconds) to meet at **half the distance**. So, **Rooney** will cross with:

Look at it another way:

When **Rooney** leaves in the **last** place, there will be **10** of his **team-mates** traveling in the **same direction** to the opposite sideline, and **other 10 teammates returning** from there.

So when **Rooney** gets to the other side, he will necessarily have **met them all**, the 10 ones that are now returning, and the 10 that are now traveling in the same direction, who he will find them returning to the initial sideline as time goes by.

We can also solve the problem by a **graphic**, drawing the **position** of every player according to the initial sideline in every moment of his way. As you can see in the graphic below, the **route** of **Rooney** intersects the way of **20 team-mates** (including **Crouch**, who meets on second 0:00).

This is a new exercise in **logical reasoning**, in which our brain focuses on numeric variables (10 seconds, 1 player / second) and **desestimates** other apparently non-numeric **fundamental variables**: the **speed** and the **direction** of **Rooney**'s movement.

This is a **football version** of the mathematical problem initially exposed by **Charles-Ange Laisant** on the **ships** that **crossed** in the path between **Le Havre** and **New York**, and later edited by the Brazilian recreational-mathematics writer **Malba Tahan**.

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