# Piles of matches

Ana emptied a matchbox on the table, distributing them in three different piles.

In those piles there were a total of 48 matches and he observed the following: “If from the first pile, as many matches as there were initially in the second and then from the second step to the third, as many matches as there were in this third pile and then, from the third a lot happened to the first as many matches as there was at that time in the first, at the end of this process the three piles will be the same ”.

How many matches did he have at first in the first pile?

Extracted from the problemate.blogspot.com.es page.

#### Solution

As in the end the three heaps remain the same, we know that in each of them there will be 16 matches. We will proceed to a kind of turning back to undo the whole process starting from the initial state.

Undoing the last step is to leave the first pile with 8 matches and pass the other eight to the third pile, which happens to be 24.

The step before this is done between piles two and three, removing half of the 24 to the third, which are 12, and leaving them in the second pile, which happens to be 28.

Finally, the first step is undone by taking half of the matches from the second pile, which is now 28 and becomes 14, and putting them in the first pile, which goes from 8 to 22.

In summary, that at the beginning we have a first pile with 22 matches, a second with 14, and a third heap with 12.