The Chosen Monks

In a secluded monastery high in the himalayan mountains lives a brotherhood of monks.

These monks spend most of their time meditating in their own room but gather once a day for lunch around a large round table. They are bound by a vow of silence that forbids them to communicate in any meaningful way with each other.

Once every 10 years, some of them are chosen by the gods for a higher purpose. During the night, the gods mark the chosen ones with a wide and bright red dot on their forehead for everyone else to see. Monks are unable to see their own forehead and must determine whether they have been chosen by meditation and reflexion alone. When the gods have made their choice and marked the chosen ones, the bells of the monastery ring loudly for one full day. The bells only ring if at least one of the monks has been chosen and marked.

As soon as a monk knows he has been chosen, he must quietly leave the monastery during the night, to follow the mountain trail that will lead him to enlightenment.

Question: Assuming that the monks are all perfectly trained in the ways of the mind and can therefore make all available deductions from the informations at hand, what happens after the bells of the monastery ring?

A few points to make the problem clearer:

• On the morning when the bells ring, all monks know that at least one of them has been marked.
• When the monks meet, once a day for lunch, they can see all other monks, know which have been chosen, and also which have left the monastery during the previous night, if any.
• The problem is to determine if, how and when the chosen monks can know that they have been chosen.

The Smarties' Great Escape

Consider a checkerboard that extends infinitely towards the up and right directions. On the bottom-left cell (that we'll call the origin), place a single smarties.

You are allowed to perform repeatedly the following action:

• if there is a smarties on a cell and the up and right neighboring cells are empty, you can remove the smarties from the considered cell, and add a new smarties on each of the up and right neighboring cells, as illustrated by the following figure.

The transformation rule.

Question: Is it possible, from the initial starting position with a single smarties in the origin cell, to apply the transformation rule so that the 6 cells corresponding to the first 3 diagonals are completely empty.

The goal is to remove all smarties from the white cells.

The following figure shows a sequence of consecutive valid transformations from the initial configuration: