\section*{Task 2 - 291022}

Two players have come up with the following game:

On a game board, 8 fields are arranged in a circle, one of these fields is designated as the starting field A. Each player has a game piece and places it on field A.

Then each player rolls a die.

If both players roll different numbers, the player with the higher number moves their piece two steps clockwise, and the other player moves their piece one step.

This movement of both pieces is considered one move.

If both players roll the same number, no move is made, and they roll again. Due to the circular arrangement of the fields, it is possible for a piece to reach or pass field A (and thus start a new lap).

The game is over as soon as after performing a move, one player's piece lands exactly on field A.

This player wins if the other player's piece is not on A.

However, if both pieces are on A, the game ends in a draw.

What is the smallest possible number of moves in which a game can end in a draw?

Justify your answer!
I. If a game ends in a draw after \(n\) "moves", the sum of the total number of steps taken by both stones is 3n. Furthermore, then one player's stone has completed a number \(a\) of full laps and the other player's stone has completed a number \(b\) of full laps, so it follows that

\[
s=8 a+8 b=8(a+b)
\]

Consequently, \(a\) is an integer multiple of the least common multiple 24 of the numbers 3 and 8; i.e., \(s=24 g\) with an integer \(g\). Therefore, \(3 n=24 g,=8 g\).

If \(g=1, n=8\), then \(8(a+b)=s=24, a+b=3\), which with positive integers \(a, b\) leads to \(a=1, b=2\) or \(a=2, b=1\); i.e., one player's stone would have completed exactly one full lap, which implies that in each of these 8 "moves" it would have to take only one step, while the other player's stone would always take two steps.

This leads to the contradiction that the latter's stone would have landed on field A by the fourth move.

Thus, \(g \geq 2\) must hold, i.e.: A draw cannot occur in fewer than 16 "moves".

II. A draw can occur in 16 "moves", for example, as follows (the fields after A are numbered \(1, \ldots, 7\)):

\begin{tabular}{l|cccccccccccccccc} 
Move & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 \\
\hline Player 1 Step Count & 1 & 2 & 2 & 2 & 2 & 2 & 1 & 1 & 2 & 2 & 1 & 1 & 1 & 1 & 1 & 2 \\
reached field & 1 & 3 & 5 & 7 & 1 & 3 & 4 & 5 & 7 & 1 & 2 & 3 & 4 & 5 & 6 & A \\
\hline Player 2 Step Count & 2 & 1 & 1 & 1 & 1 & 1 & 2 & 2 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 1 \\
reached field & 2 & 3 & 4 & 5 & 6 & 7 & 1 & 3 & 4 & 5 & 7 & 1 & 3 & 5 & 7 & A
\end{tabular}

With I. and II., it is proven: The smallest possible number of "moves" in which a draw can occur is 16.