## Access Type

Open Access

## Entry Number

121

## Start Date

4-5-2017 10:15 AM

## End Date

4-5-2017 10:30 AM

## Department

Mathematics

## Abstract

Abstract: Assume there is a game that takes place on a 20x20 checkerboard in which each of the 400 squares are filled with either a penny, nickel, dime, or quarter. The coins are placed randomly onto the squares, and there are to be 100 of each of the coins on the board. To begin the game, 59 coins are removed at random. The goal of the game is to remove each remaining coin from the board according to the following rules: 1. A penny can only be removed if all 4 adjacent squares are empty. That is, a penny cannot be removed if there is a coin to the left, right, above, or below the penny. 2. A nickel can be removed if at least 3 of the 4 adjacent squares are empty. 3. A dime can be removed if at least 2 of the 4 adjacent squares are empty. 4. A quarter can be removed if one or more of the adjacent squares are empty. We will show that the game cannot be won if 59 coins are initially removed. To prove this, an understanding of monovariants and invariants is required. A monovariant is a value that only changes in one direction; it either always increases or decreases at each step. An invariant is a value that never changes or no longer changes after a point in the process. The rules of the game will limit how the total dimensions of empty squares, such as area and perimeter, are changing. To show that the game cannot be won when initially removing 59 coins, we must show that the changing dimensions cannot equal the dimensions of the final winning board.

## Faculty Mentor(s)

Michael Coco

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#### Included in

A Game of Monovariants on a Checkerboard

Abstract: Assume there is a game that takes place on a 20x20 checkerboard in which each of the 400 squares are filled with either a penny, nickel, dime, or quarter. The coins are placed randomly onto the squares, and there are to be 100 of each of the coins on the board. To begin the game, 59 coins are removed at random. The goal of the game is to remove each remaining coin from the board according to the following rules: 1. A penny can only be removed if all 4 adjacent squares are empty. That is, a penny cannot be removed if there is a coin to the left, right, above, or below the penny. 2. A nickel can be removed if at least 3 of the 4 adjacent squares are empty. 3. A dime can be removed if at least 2 of the 4 adjacent squares are empty. 4. A quarter can be removed if one or more of the adjacent squares are empty. We will show that the game cannot be won if 59 coins are initially removed. To prove this, an understanding of monovariants and invariants is required. A monovariant is a value that only changes in one direction; it either always increases or decreases at each step. An invariant is a value that never changes or no longer changes after a point in the process. The rules of the game will limit how the total dimensions of empty squares, such as area and perimeter, are changing. To show that the game cannot be won when initially removing 59 coins, we must show that the changing dimensions cannot equal the dimensions of the final winning board.