The basket of balls

Versión española: Contrarreloj ciclista

Before the final of the 2012 European Championship, the Spanish team is practicing penalty shots.

The assitants pick all the balls and put them into a basket beside the penalty spot.

Xabi Alonso is the first to train his shots, and he uses half of the balls in the basket plus one ball more.

Fernando Torres goes in second place, and he also uses half of the remaining balls into the basket, plus one ball more.

Afterwards, Cesc tries kicking, and he uses half of the reamining balls plus one.

Then, Iniesta does the same, and he also uses one half of the balls in the basket plus one.

Finally it's up to Sergio Ramos, but he can not kick any ball as there's no ball in the basket.

How many balls where there in the beginning?

To solve it the best possible way, we start counting from the last player to kick.

We know that Sergio Ramos found no ball in the basket, so Iniesta necessarily kicked 2 balls. That way he took half of the balls (2/2=1) plus 1.

We can calculate it as follows: we take the balls he leaves in the basket (0), we add 1, and multiply the result by two:
 

(0+1) · 2 = 2

When Fabregas takes half the balls remaining in the basket there must be the 2 balls Iniesta kicked later and the balls Cesc used. That is, there were 6 balls: he took half the balls (6/2=3) plus 1, and left the 2 balls Iniesta kicked. We can calculate it as in the previous case:
 

(2+1) · 2 = 6

Following the same scheme, Fernando Torres will find 14 balls: he picks half of them (7) plus one, and leaves 6 to Cesc. If we calculate backwards:
 

(6+1) · 2 = 14

And finally we come to Xabi Alonso. Following the reasoning, he will initially have:
 

(14+1) · 2 = 30 balls

And this is the correct answer. In the beginning, we got 30 balls in the basket!

= 30 balls