Parity , odd and even

The most basic split of whole numbers is into even ($0, 2, 4, 6, \dots$) and odd ($1, 3, 5, 7, \dots$). Tiny as this split seems, it cracks open many puzzles.

Idea

A whole number is even if it can be written as $2k$ for some integer $k$, and odd if it is one more than an even number, $2k + 1$.

Sum rules (try with examples to convince yourself):

• Even $+$ Even $=$ Even. (e.g., $2+4=6$.)
• Odd $+$ Odd $=$ Even. (e.g., $3+5=8$.)
• Even $+$ Odd $=$ Odd. (e.g., $2+3=5$.)

Product rules:

• Even $\times$ anything $=$ Even.
• Odd $\times$ Odd $=$ Odd. (e.g., $3 \times 5 = 15$.)

Why parity is powerful. Many puzzles have a hidden invariant that depends only on parity. If the puzzle says "the sum of $X$ items always changes by an even amount in each move", and the start and end have different parities, the puzzle is impossible , regardless of how clever you are.

Quick test. A number is even iff its last digit is $0, 2, 4, 6, 8$. The other digits don't matter. So $1{,}23{,}45{,}678$ is even and $98{,}76{,}54{,}321$ is odd.

Worked examples

Example 1. Is $9 \times 11 + 2$ even or odd?

$9 \times 11$ is odd $\times$ odd $=$ odd. Adding $2$ (even) gives odd. So the result is odd. (Indeed $99 + 2 = 101$.)

Example 2. Show that the sum of two consecutive numbers is always odd.

Consecutive numbers are $n$ and $n+1$ , one even, one odd. Even $+$ Odd $=$ Odd.

Example 3. Find all pairs of two-digit numbers $a, b$ such that $a + b = 100$ and both are odd.

If both are odd, $a + b$ is even , so $100$ being even is consistent. Many examples: $1 + 99$, $3 + 97$, $11 + 89$, etc. The constraint of being odd just means both end in $1, 3, 5, 7, 9$.

Example 4. $10$ coins lie on a table, all showing heads. In one move you flip exactly two coins. Can you reach the state where all show tails?

Each move flips two coins , the number of heads changes by $0$ (if both already same) or by $\pm 2$. So heads-count remains even. You start with $10$ heads (even) and want $0$ heads (even). Parity allows it , and indeed it can be done in $5$ moves.

Try it yourself

1. Is the sum of an even and an odd number always odd? Justify.
2. Is $7 + 8 + 9 + 10$ even or odd? Why?
3. Find two odd numbers whose sum is $20$.
4. Is $13 \times 15 \times 17$ even or odd?
5. Without computing, is $1234 \times 567$ even or odd?
6. Show: if you add three odd numbers, the sum is always odd.
7. A grid has $3$ rows and $3$ columns. Each cell is $+1$ or $-1$. Can you arrange them so that the product of each row, column equals $-1$?
8. The sum $1 + 2 + 3 + \dots + 100 = 5050$. Even or odd? Why?

Activity

Play "heads-tails": $10$ coins all heads up. Take turns flipping any two coins. Try to reach all tails. Discuss why you can or cannot.

Pick any $5$ consecutive numbers. Add them. Is the sum always divisible by $5$? Test for a few starts.