SIX CHILDREN ARE EVENLY SPACED AROUND A CIRCULAR TABLE. INITIALLY,...

5. Six children are evenly spaced around a circular table. Initially, one has a pile

of n > 0 sweets in front of them, and the others have nothing. If a child has

at least four sweets in front of them, they may perform the following move:

eat one sweet and give one sweet to each of their immediate neighbours and

to the child directly opposite them. An arrangement is called perfect if there

is a sequence of moves which results in each child having the same number

of sweets in front of them. For which values of n is the initial arrangement

perfect?