Some SASMO problems look like magic tricks: the answer seems impossible to reach, until you realize something never changes. That unchanging something is called an invariant. It is a secret key that unlocks the hardest questions. Master this concept, and you will turn confusing puzzles into simple confirmations.

What Exactly Is an Invariant?

An invariant is something that does not change during a process. Think of a game where you swap two numbers on a board. The sum of all numbers stays the same. That sum is an invariant.

Invariants come in many forms: the parity (even or odd) of a set, the remainder when divided by a number, the total weight of objects, or the color pattern of a chessboard. Once you identify the invariant, you can compare the starting state with the final state. If they don’t match, the final state is impossible.

For SASMO, invariants are most common in combinatorics and number theory problems. But they also show up in geometry and algebra. Learning to spot them is a skill you build over time.

Why Invariants Matter in SASMO

• They turn open ended questions into yes/no checks.
• They eliminate impossible answers quickly.
• They require no guesswork: the invariant tells you the truth.
• They work beautifully with modular arithmetic and parity.
• They impress judges with clean, logical reasoning.

For example, a problem might ask: “Can you tile a 6×6 board with 2×1 dominoes after removing opposite corners?” The invariant is the color count. A standard chessboard has 32 black and 32 white squares. Removing opposite corners removes two squares of the same color. Now you have 30 of one color and 32 of the other. Each domino covers one black and one white. So tiling is impossible. This invariant saves you from hours of trial and error.

How to Spot an Invariant in a Problem

Follow these steps to find the invariant that cracks the problem.

1. Read the process carefully. Understand each allowed move. Write down what changes and what might stay the same.
2. Test small examples. Perform a few moves on a small version of the problem. Look for a quantity that remains constant. It could be a sum, difference, product, remainder, or count.
3. Think about parity. The odd/even property of a total or a difference is the most common invariant. If the problem involves flipping signs, swapping numbers, or moving pieces, check parity first.
4. Consider modular arithmetic. If numbers are involved, try modulo 2, 3, or another small number. Often the invariant is the remainder of a sum or product.
5. Check for colorings or bipartite properties. For tiling, path, or matching problems, assign colors to positions. The number of each color can be an invariant.
6. Verify the invariant against the goal. Once you think you found it, compute its value at the start and at the target. If they differ, the target is unreachable. If they match, the target might be possible, but you need to show it explicitly.

Common Invariant Techniques for SASMO

Here is a table of techniques, examples, and frequent mistakes.

A monovariant is a related idea: it always increases or always decreases. While an invariant stays fixed, a monovariant can help you prove a process must stop. For SASMO, invariants are more common, but monovariants appear occasionally.

Worked Example: The Card Swap Game

Let’s solve a problem typical for SASMO Grade 7 or 8.

“Alice writes the numbers 1, 2, 3, 4, 5, 6 on a board. Each move, she picks two numbers, erases them, and writes their sum. She continues until only one number remains. What are the possible values of that final number?”

Step 1: Find the invariant. The sum of all numbers on the board is always the same. Initially, the sum is 1+2+3+4+5+6 = 21. When you erase two numbers, say a and b, and write a+b, the total sum becomes old sum minus a minus b plus (a+b) = old sum. So the total sum is invariant.

Step 2: Apply to the final state. At the end, only one number is left. That number must equal the invariant total, 21. So the only possible final number is 21.

Step 3: Check reachability. Can she actually get to 21? Yes. For example, combine 1 and 2 to get 3, then combine that 3 with the existing 3 to get 6, combine 4 and 5 to get 9, then combine the 6 and 9 to get 15, and finally combine 15 and the remaining 6 (from the original 6) to get 21. The invariant proves that no other number is possible.

This problem is a classic SASMO style. Without the invariant, you might try all possible sequences and get lost. With it, the answer is obvious.

Advice from the Pros

“The hardest part is not solving the problem. It is believing that you can find an invariant. Start small. Play with the numbers. The invariant will show itself. And once you see it, the problem breaks open.” – seasoned SASMO coach

That advice is spot on. Many students freeze because they think invariants are mysterious. But they are just careful observation. Practice with simple games, like the coin flipping puzzle or the chocolate bar breaking problem, and you will train your brain to look for what stays the same.

Your Turn: Practice Problems

Now it is time to test your skill. Try these problems and look for the invariant.

• A 5×5 grid has 25 squares. You remove the four corners. Can you tile the remaining shape with 2×1 dominoes? (Hint: color the grid in a checkerboard pattern.)
• You have 10 piles of stones. In one move, you take one stone from each of two piles and add them to a third pile. Is it possible to end with all stones in one pile? (Hint: track the parity of the number of piles with odd counts.)
• Start with the numbers 1, 3, 5, 7, 9. In each move, replace two numbers a and b with |a – b|. Can you end with a single 0? (Hint: the sum modulo 2 is invariant, but also check the sum modulo something else.)

For more practice, check out our grade by grade SASMO problem sets. They include problems that specifically train invariants.

Building Your Invariant Intuition

You cannot master invariants overnight. But you can build intuition through deliberate practice. Each time you solve a problem, ask yourself: “What stayed the same? Could I have used a different invariant?” Over time, you will see patterns.

Also combine invariants with other strategies. For example, reading the problem carefully to avoid misinterpreting moves is critical. See our guide on reading SASMO questions carefully to sharpen your first step.

Another powerful pairing is invariants with modular arithmetic. Many SASMO number theory problems use modulo invariants. Our article on 5 essential properties of integers will deepen your understanding.

Final Push: Make Invariants Your Go To Tool

The next time you face a SASMO problem that seems to have too many possibilities, stop. Do not start guessing. Look for the invariant. It might be hiding in plain sight: the sum, the parity, the modulo remainder, or the color count. Once you find it, the problem shrinks from a universe of options to a single point.

Start training today. Grab a practice set, find a partner, and challenge each other to spot invariants before solving. The more you do it, the more natural it becomes. You will find yourself saying, “Oh, that can’t happen because of the invariant,” before you even finish reading the question.

Good luck, and remember: every SASMO champion once learned to see what never changes. You are next.