Booth Id:
MATH048
Category:
Mathematics
Year:
2018
Finalist Names:
Jarviniemi, Olli (School: Valkeakosken Tietotien Lukio)
Abstract:
\textit{Draw of choice} (Finnish: "hutunkeitto") in Finnish baseball can be seen as a finite two-player game with winning and losing positions. By inspecting winning and losing positions one can determine the winning moves if such exist.
Let the players be $A$ and $B$. \textit{Draw of choice} can be stated as follows: let positive reals $x, a_1, a_2, b_1, b_2$ be given. $A$ begins and subtracts some number from the interval $[a_1, a_2]$ from $x$. After this $B$ subtracts some number from the interval $[b_1, b_2]$ from the current number. This procedure is continued until the number is negative or $0$. The player to make the last move wins.
Following results are found: let $k \geq 0$ be an integer. If $x$ satisfies $k(a_1 + b_2) < x \le (k+1)a_2 + kb_1$ $A$ wins. $A$ loses if $(k+1)a_2 + kb_1 < x \le (k+1)(a_1 + b_2)$. Similar results hold for $B$. Also, if $a_2 - a_1 > b_2 - b_1$, then $A$ wins for all sufficiently large $x$.
The solution is computationally efficient: determining the winner for a single value $x$ can be done by finding the integer for which $k(a_1 + b_2) < x \le (k+1)(a_1 + b_2)$ holds. After this it should be checked if $x \le (k+1)a_2 + kb_1$ holds. If yes, $A$ wins, and otherwise $A$ loses.