Investigation Question $1$

A television show sets up a puzzle for its contestants to try to solve.

Some buried treasure is hidden on an island. The treasure is hidden in a treasure chest. Each contestant stands by one of the treasure chests.

The treasure is hidden according to the following rule:

It is not hidden in chest $1.$

Chest $2$ is left empty for the time being.

It is not hidden in chest $3.$

Chest $4$ is left empty for the time being.

It is not hidden in chest $5.$

The pattern of crossing out the first chest and then alternate chests is continued until only one chest is left. This will involve going round the circle several times, continuing the pattern. The treasure is hidden in the last chest left.

The diagrams below show how the last chest is chosen

$=>$ After one round, chests $1,3,5,7,9$ and $11$ have been discounted.

$$ After the second round, chests $2,6$ and $10$ have also been discounted.

$=>$ After the third round, chests $4$ and $12$ have also been discounted. This leaves only chest $8.$

The treasure is therefore hidden in chest $8.$

Unfortunately for participants, the number of contestants changes each time, as does the number of chests.