Consider a matchstick game where two players take turns removing $1,2,3,$ or $4$

matches from a pile of $n$ matches. The winner is the player who removes the last

match. In the context of algorithm design, which type of problem does this

matchstick game represent if the game is played optimally?

SOLUTION

If the initial number of matches, $n,$ is not a multiple of $5,$ the first player can win if

they play optimally $($see the Optimal Strategy below$).$

If $n$ is a multiple of $5,$ the first player will lose if the second player plays optimally

$($the Optimal Strategy below$).$ In this case, the player can pick any number of

matches.

Optimal Strategy:

The player aims to ensure that the remaining number of matches after their move is

a multiple of $5.$ So$,$ if the current number of matches, $n,$ is not a multiple of $5,$ the

player removes nmod$5$ matches in each turn.

Repeating the Strategy:

The player repeats the Optimal Strategy throughout the game to maintain control

and assure their victory.

Example$1$:

Assume there are initially $11$ matches on the table. The game proceeds as follows:

Player $1$ removes $1$ match, leaving $10$ matches.

Player $2$ removes $1$ match, leaving $9$ matches.

$($Note: If Player $2$ removed $2$ matches, then Player $1$ should remove $3$

matches in the next step. If Player $2$ removed $3$ matches, then Player $1$

should remove $2$ matches in the next step. If Player $2$ removed $4$

matches, then Player $1$ should remove $1$ match in the next step.$)$

Player $1$ removes $4$ matches, leaving $5$ matches.

Player $2$ removes $2$ matches, leaving $3$ matches.

$($Note: Player $2$ can remove any number of matches they want. It

doesn't matter at this point, as Player $1$ can scoop up what is left and

win.$)$

Player $1$ removes $3$ matches and wins the game.

HINT: The magnitude of $n,$ defined as the number of digits in its decimal

representation, serves as a measure of the problem's size.

An example of a "Decrease by a Constant Factor" problem.

An example of a "Decrease by Constant" problem.

An example of a "Variable Size Decrease" problem.

This problem does not fit any of the mentioned problem types.