2.4 Applications of Logic in Puzzles and Games
Key Takeaways
- Puzzles and games test whether you can organize conditions before calculating; rushing to arithmetic is often the wrong first move.
- A clear table, grid, Venn diagram, or case list turns scattered clues into visible restrictions and prevents repeated counting.
- When a puzzle asks what must be true, eliminate possibilities using the given rules only; do not add assumptions from ordinary life.
- For written response, explain the strategy and conclusion in order so the marker can follow how each clue narrowed the cases.
Turning Clues Into Structure
The Mathematics 30-2 bulletin notes that students typically do well on puzzles and games involving numerical and logical reasoning. That does not mean these questions are automatic. They reward students who slow down long enough to organize the clues, identify restrictions, and check that every rule has been used. The setting may look playful, but the skill being tested is precise mathematical communication.
A puzzle or game question usually gives a small system: players, positions, cards, colours, scores, paths, or categories. Your job is to decide what must be true, what could be true, or how many arrangements are possible. The best first step is not a formula. It is choosing a representation.
Pick the right organizer
| Clue type | Useful organizer | Why it works |
|---|---|---|
| People matched to roles | Logic grid or table | Tracks yes/no possibilities |
| Movement on a path | Diagram or ordered list | Preserves sequence |
| Categories with overlap | Venn diagram | Separates both, only, neither |
| Repeated choices | Tree diagram or slots | Shows cases without skipping |
| Scores or rankings | Table with inequalities | Keeps comparisons visible |
For machine-scored questions, the organizer may stay on scrap paper. For written response, describe the organizer in words if the final answer depends on it. The official instructions allow scrap paper, but work on scrap paper is not marked; your written-response answer must present the reasoning that earns credit.
Worked puzzle example
Three students, Ali, Bree, and Cole, each choose one game: chess, cards, or darts. Each game is chosen by exactly one student. The clues are:
- Ali did not choose chess.
- Bree chose a game with five letters.
- Cole did not choose darts.
Start with the most restrictive clue. The five-letter games are chess and darts. Bree chose chess or darts. Ali did not choose chess. Cole did not choose darts.
Make cases for Bree. If Bree chose chess, then Ali cannot choose chess, so Ali must choose darts or cards. Cole cannot choose darts and chess is used, so Cole must choose cards. Then Ali must choose darts. This case works.
If Bree chose darts, then Cole cannot choose darts, and Ali cannot choose chess. The remaining games for Ali and Cole are chess and cards; Ali cannot have chess, so Ali gets cards and Cole gets chess. This case also works.
Therefore the clues do not determine one unique assignment. What must be true? Bree did not choose cards. What could be true? Ali could choose darts or cards, depending on the case. This distinction between must and could is a common trap.
Game-rule reasoning
Many game questions use if-then rules. Example: "A player may move to the centre only after collecting a key. A player who lands on a penalty square loses a turn." If the question says Nina moved to the centre, you can conclude she collected a key only if the rule is written as a requirement: moving to the centre is allowed only after collecting a key. If it says Nina collected a key, you cannot conclude that she moved to the centre; the key may simply make the move possible.
Words such as only if, unless, and must matter. "A move to the centre is allowed only if a key was collected" means key collection is necessary for the centre move. "A player may move to the centre if a key was collected" says the key is sufficient to allow the move, but the player may choose not to move.
Casework without losing cases
If a puzzle has alternatives, use organized casework. Name each case by the decision that creates it, then test the remaining clues. Do not erase failed cases without noting why they failed. A failed case can be valuable evidence in a written explanation.
For example, suppose a four-card code uses one shape and one colour on each card. The rules say exactly one card is red, the triangle is not blue, and the circle is red or green. You might split into Case 1: circle red and Case 2: circle green. In Case 1, the only red card is settled, so all other cards are not red. In Case 2, the red card must be a different shape. That split prevents you from accidentally allowing two red cards later.
Numerical puzzles and invariants
Some games ask about a quantity that changes after each move. Look for an invariant, something that stays the same. If every move adds 2 points, the parity of the score stays the same: an even score remains even and an odd score remains odd. If every move swaps two tokens, the number of tokens may stay fixed even though positions change.
Example: A token starts on square 1. Each move adds 4 to the square number. Can the token land on square 18? The reachable squares are 1, 5, 9, 13, 17, 21, and so on. They all leave a remainder of 1 when divided by 4. Square 18 leaves a remainder of 2, so it is not reachable. This is deductive reasoning based on a pattern rule, not just a list of tries.
Final verification
Before choosing an answer, test it against every clue. For a must-be-true question, also ask whether a second valid arrangement would make the statement false. For a could-be-true question, one complete valid arrangement is enough. For a cannot-be-true question, show which clue is contradicted.
In written response, end with a sentence that matches the command word: "Therefore Bree must have chosen either chess or darts, so Bree cannot have chosen cards" or "This arrangement satisfies all three clues, so the statement is possible." That kind of conclusion turns puzzle work into a complete Math 30-2 answer.
A puzzle has four players matched to four different roles. The clues mostly say which player cannot have which role. Which organizer is usually most efficient?
A game rule says, "A player can enter the bonus room only if the player has a silver key." Taylor entered the bonus room. What must be true?
A token starts at 2 and each legal move adds 6. Which square cannot be reached?