# CTK Insights

• ## Pages

21 May

### Parity Games

Parity is the simplest mathematical concept after counting. 1 is an odd number, 2 is even, and then they come intermittently: 3, 5, 7, ... are odd, 4, 6, 8, ... are even. A pile of an even (but not odd) number of items can be divided into two piles of equal sizes. An odd number is one more than an even number. It follows that, since the sum of two even numbers is even, so is the sum of two odd numbers. On the other hand, the sum of an odd and an even number is always odd. When you start counting by twos from an odd number, the numbers in the sequence you come with are all odd.

The activities below, suggest additional properties of odd and even numbers and offer a playful reinforcement of the parity related concepts.

• ### Plus or Minus Game

Given a sequence of (positive) integers, the task is to place either "+" or "-" between every pair of two neighbors in the sequence. It could be a game for one or two players. One wins when the resulting number is odd, the other when it is even. The first player has the right to pass the first move.

In the applet, the numbers displayed are consecutive integers, but it need not be so. The game could be played many times over on a piece of paper.

• ### Squares and Circles

The game starts with a sequence of circles and squares. A move consists in selecting any two shapes. The two shapes if identical are replaced with a square. Different shapes are replaced with a circle. This game, too, can be played by one or two players. One player should aspire to leave a circle as a last shape; in a two players game the first one wins if circle is the last shape. The first player can choose to skip the first move.

• ### White and Black Balls in Urn. 1 in, 2 out. What Color Remains?

This is a puzzle to ponder. In a large urn there are 75 white balls and 150 black ones, and beside the urn is a big pile of black balls. Now, the following two-step operation is performed repeatedly. First, two balls are withdrawn at random from the urn and then

1. if they are both black, one of them is put back and the other is thrown away,
if one is black and the other white, the white one is put back and the black one is thrown away,

2. if they are both white, they are both thrown away and a black ball from the pile is put into the urn.

Therefore, whatever the case, at each stage two balls removed from the urn and one is put back, thus reducing the number of balls in the urn by one. Eventually, then, the urn will reach the point of containing just a single ball. The question is "What color is this last ball?"

Try replacing "75" and "100" with smaller numbers.

• ### Solitaire in Two Colors

Numbers from 1 through N are drawn in red, the next N are drawn in blue. The red and blue cards are reshuffled independently and then intermixed: one red card followed by one blue card followed one red card, and so on. The cards are placed in a circle. Two cards can be removed in adjacent pairs provided the numbers on the two neighbors are of the same parity: either both odd or both even. The process continues until no two neighbors are of the same parity.

It may be a surprise (that requires an explanation) that if N is even all card will be eventually removed, regardless of the sequence of moves.

• ### Solitaire on a Circle

A puzzle below is played with a number of counters arranged in a circular pattern. One of the faces of each counter is red, the other is green. A move consists in clicking on a green counter. When you do, the counter is removed. At the same time, its neighbors (if any but at most 2 of course) are flipped over. The goal is to remove all the counters. Beware though that the puzzle is not always solvable. There is a simple condition to check in advance whether it is.

The puzzle is solvable only if the number of green counters in the initial setup is even and nonzero.

• ### Farmer and Wife To Catch Rooster and Hen

This is a puzzle by the famous Sam Loyd. On a chessboard, there are pieces that represent Farmer, his Wife, Rooster and Hen. Each may only move either vertically or horizontally to an adjacent square. Farmer and Wife catch a chicken whenever they step into the square with the chicken. Farmer and Wife move first (i.e., each steps into an adjacent square) and then Rooster and Hen move trying to escape from the peasant couple. Can they?

The applet implementation generalizes Sam Loyd's puzzle in two ways: first, the board is not necessarily 8×8; second, the applet allows for any combination of single opponents from each pair.