Question 83
It is Ann's birthday. She, her brother Jim, and 8 of her friends sit around a table in the shape of a regular decagon (a 10-sided polygon). Jim sits directly opposite Ann at the table. The 10 children play a game called *throw the parcel*, which works as follows. At the beginning, Ann has the parcel. The game then consists of $n$ moves. In each move, the person who has the parcel throws it to a person sitting immediately beside them (their neighbour), or to a person sitting immediately beside their neighbour. Whoever has the parcel at the end wins the game.
Before the game begins, Ann says to Jim, “there are over a million more ways for the parcel to travel to end up with me than with you”.
Determine, with proof, the values of $n$ for which Ann's claim is correct.
Before the game begins, Ann says to Jim, “there are over a million more ways for the parcel to travel to end up with me than with you”.
Determine, with proof, the values of $n$ for which Ann's claim is correct.