In one sentence
Sort the integers into just two kinds: odd and even. Throw away every other piece of information.
The pay-off: many complicated operations change nothing at all when seen through these “two-color glasses”. If some quantity's parity never changes, and the goal has a different parity from the start, then the goal can never be reached. You do not have to try every possible way.
The rule you will use most: when you add up a list of integers, the parity of the sum depends only on how many odd numbers there are. An odd number of odd numbers gives an odd sum; an even number of odd numbers gives an even sum.
Predict first
Without working it out: is odd or even?
From to there are odd numbers: . An odd number of odd numbers adds up to an odd number, and adding even numbers does not change the parity. So the sum is odd. (If you do calculate it, it is .)
Stop 1 · Parity arguments (Vol. 6)
Put a or a in front of each of . Can you make the result equal ? Try a few times yourself first.
Example: can ±1 ± 2 ± 3 ± … ± 10 equal 0?
Step 1 / 5- All plus signs
Start with everywhere: the sum is , an odd number.
Stop 2 · Integer solutions of equations (Vol. 2)
and differ by , so they have the same parity. Whenever this pair turns up, think parity.
Predict first
and are integers. Can equal ?
The two factors have the same parity. If both are odd, the product is odd; if both are even, the product is a multiple of . is even but not a multiple of , so neither case works.
Example: does x² − y² = 2026 have integer solutions?
Step 1 / 5- Factor first
By the difference of squares, the left side is a product of two integers.
Here parity (mod ) came first; it was not quite enough, so we looked one step further: “is it a multiple of ?” (mod ). A parity argument is the simplest case of choosing a modulus. The Choosing a Good Modulus line is the upgraded version of the same idea.
Stop 3 · Coloring (Vol. 7)
A checkerboard is parity drawn as a picture: color the cell in row , column by whether is odd or even. Moving to a neighboring cell changes by , so the color always changes.
A board has one beetle in every cell. At the whistle, every beetle crawls to a neighboring cell (one that shares a side). Must some cell end up empty?
Example: 25 beetles move house at once
Step 1 / 5- A first thought
beetles, cells. With a clever plan, maybe they can all just swap places?
On a board there are cells of each color, and this argument breaks down. In fact, it is then possible to leave no cell empty (just let neighboring beetles swap places in pairs).
Stop 4 · Operations and games (Vol. 7)
Back to the cups at the top of the page: cups mouth up, and each move flips exactly of them.
Example: 7 cups, flip 4 at a time: can they all face down?
Step 1 / 5- Watch one number
Ignore which cups are which; just count how many cups face down. At the start it is .
Change one number and the conclusion changes. Below, you flip cups at a time: the change is odd, so the parity switches on every move. Can it be done this time? In how few moves?
Practice
What is the remainder when is divided by ? (Enter or ; do not work out the sum.)
The numbers are written on a board. Each time, you erase any two numbers and and write their difference . After doing this times only one number is left. Is it odd or even?
Of the four numbers , exactly one cannot be written as a difference of two integer squares . Which one?