Two-color the board and count what each move does to the imbalance. If it can't fix the imbalance, the goal is impossible.
Method · Parity
Intro
Parity is the most useful invariant in combinatorics because it’s usually staring at you. Color the board, label even-vs-odd, bipartition the graph — pick the right 2-coloring and most reachability and tiling problems collapse to a one-line counting check. The move flips the color (or doesn’t); after $n$ moves the parity is forced; compare to the goal and you’re done.
✓ Intro · expand
Try first (productive failure)
Before the worked example: spend 60 seconds taking your best shot at this.
A guess is fine — being briefly wrong about a problem makes the explanation
land harder when you read it. This appears once per tutorial; skip
if you already know the trick.
60s
✓ Try first · expand
Worked example
An 8×8 chessboard has its two opposite corners removed (the squares at (1,1) and (8,8)), leaving 62 squares. Can the remaining region be tiled by 1×2 dominoes? Answer 1 for yes, 0 for no.
✓ Worked example · expand
Practice 1 of 3Type a fraction, decimal, or expression — mathjs parses it.
✓ Practice · expand
Reflection
When you read a problem like “can this state reach that state?” or “can this region be tiled?”, what's the cue that tells you to reach for a 2-coloring? And when parity says “yes” (start and goal agree), why isn’t that enough — what extra work is still on you?