Invariants & Parity Arguments

• What an invariant is: some quantity or property that never changes no matter how a process is repeated, and how spotting one instantly solves "can you reach this state?" puzzles.
• How parity (odd/even) is the simplest and most common invariant in olympiad puzzles.
• How colouring arguments (like colouring a chessboard) turn a geometric or combinatorial puzzle into a simple counting argument.

1. Invariant — a property preserved by every allowed move in a puzzle. If the target state doesn't match the invariant of the start state, the target is provably unreachable — no amount of trying will work.
2. Parity as an invariant — many operations (swap two numbers, flip two coins, add 1 to two piles) preserve the parity (odd/even-ness) of some total or difference. If the starting parity and target parity differ, it's impossible.
3. Colouring arguments — colour a grid or board in a pattern (like a chessboard) and count how many squares of each colour a shape covers; if every allowed tile always covers a fixed number of each colour, mismatched counts prove certain coverings impossible.
4. Checking small cases first — before hunting for the invariant, try the puzzle on tiny versions to notice what number stays fixed — that observation usually is the invariant.

Start with the numbers 1, 2, 3, …, 10 written on a board. In each move, erase any two numbers a and b and write |a − b| instead (so the count of numbers decreases by 1 each move). After 9 moves, one number remains. Can it be even?

Step 1 — track the SUM of all numbers on the board modulo 2 (its parity)
Step 2 — replacing a, b with |a-b| changes the sum by (a+b) - |a-b|, which is always even
          (if a≥b: (a+b)-(a-b) = 2b, an even number)
Step 3 — so the parity of the total sum never changes across any move — this is the invariant
Step 4 — initial sum = 1+2+...+10 = 55, which is odd
Step 5 — since parity is invariant, the final single number must also be odd
Answer — no, the final number can never be even; it must be odd

• Trying to solve a "can you reach this state" puzzle by brute-force search instead of first hunting for an invariant — invariants often make the answer obvious in one line.
• Picking a quantity that looks invariant but actually changes under some moves — always verify algebraically that the candidate quantity truly stays fixed (or changes in a controlled, provable way, like "always even").
• In colouring arguments, forgetting to check that the colouring pattern is applied consistently across the entire board before counting.

• A frog starts at position 0 on a number line and can jump +3 or −3 each time. Can it ever land exactly on 7? (Position always stays a multiple of 3 in terms of change from 0 — wait, position = 3k for some integer k, and 7 is not a multiple of 3, so no.)
• On an 8×8 chessboard coloured normally, if two opposite corner squares are removed (both the same colour), can the remaining 62 squares be tiled by 31 dominoes? Why or why not? (Each domino covers one black and one white square; removing two same-colour corners leaves 30 of one colour and 32 of the other, an unequal split — so tiling is impossible.)
• If you start with an even number of coins showing heads and each move flips exactly two coins, can you ever reach a state with an odd number of heads? (Flipping two coins changes the heads-count by −2, 0, or +2 — always an even change — so parity of the heads-count is invariant; starting even, it stays even forever.)

Open the Practice tab for graded questions on Invariants & Parity Arguments.

• Coin-flip invariant simulator: flip pairs of virtual coins repeatedly and watch the heads-count parity never cross from even to odd.
• Mirror / body / home activity: try the "mutilated chessboard" domino puzzle with real graph paper and dominoes cut from cardboard, and see the impossibility firsthand.
• Voice or text reflection with AI Mentor: explain why a Rubik's cube can never be solved by swapping just two stickers — corner/edge parity invariants forbid it.

• "Explain what an invariant is to a Class 6 student using a coin-flipping game with a friend."
• "What is one common mistake students make when guessing an invariant, and how would you catch yourself making it?"
• Stretch: "How does the mutilated chessboard argument connect to error-detection codes or puzzle-solvability checks in computer games?"

• Complete the core practice + one extension activity (photo, table, short reflection, or mini-project) for base XP + topic badge.
• 5-7 day streak or family discussion note = multiplier + visible artifact in parent/principal dashboard.
• Best real-world application stories (anonymised) featured on class or national leaderboard.

• One hands-on project or measurement using the Drishti kit or household items that makes the concept physical.
• Direct link to at least one Future Skill track (Money Management, Green Tech, Cyber Defenders, Micro-Entrepreneurship, AI Mastery, Sustainable Living, Personality Development).
• Coding extension where relevant (simple script, simulation, or data logging).

This material emphasises experiential "learning by doing", competency (apply/create/analyse), vocational exposure, critical thinking, and multidisciplinary connections. Designed to feed live worlds, AI Mentor (with memory), gamification, robotics, parent analytics, and future skills — not just exam prep.

Portfolio Evidence Idea: Your photo/table/reflection/project + one sentence on "How this helps me in real life or a possible future path."

Open the Practice tab for aligned questions (easy/medium/hard + case-based) with full AI scaffolding.

See curriculum for cross-links and the full future-skills/robotics chapters.