Extremely Hot Chocolate!

Discrete Mathematics Level 5

Today, Brian is generous and wants to offer a hot chocolate to all of his \(n\) friends. Unfortunately, Brian has enough money only for \(n-1\) hot chocolates, so he decided to play a game.

The game starts by placing his \(n\) friends around a table, and numbering them from 0 to \(n-1\) in clockwise order.

  • Person 0 starts off with an empty plastic cup.
  • Every person who directly receives the cup from another person around the table, wins a hot chocolate and leaves the game.
  • Once person \(p\) is gone, the cup is taken by the person to the left of \(p\).
  • The game ends when there is only one person left, who unfortunately will not receive a hot chocolate.
  • In the first move, person 0 gives the cup to person 1.

Let us denote as \(L_n\) the last person in a game of \(n\) people.

For example with 5 people the game would play out as:


So \(L_5=2\).

What is the digit sum of \(\; (L_{2^{150}-1}\mod(10^9+7)) \;\;?\)

Image Credit: Wikipedia.

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