60-second revision
- Rewrite the whole expression with real signs first; never substitute and calculate at the same time.
- BODMAS: Brackets, Of, Division and Multiplication (left to right), Addition and Subtraction (left to right).
- ÷ and × have equal priority — work them left to right. 24 ÷ 4 × 2 = 12, not 3.
- Interchange questions: test each option fully; only one swap balances the equation.
- Fill-the-signs questions: start with the option that keeps the result near the target size.
- Hidden rules (a # b) usually use a² ± b², (a + b)², a × b ± k or a × (a ± b). Check both examples.
Type 1: symbol substitution
The question redefines signs or letters (for example, "+ means ×") and asks for a value. Do it in two clean passes. Pass one: rewrite the whole expression with the real signs. Pass two: evaluate with BODMAS. Mixing the passes is the main source of errors.
Worked example: if × means +, + means −, − means ÷ and ÷ means ×, find 30 − 5 ÷ 2 × 4 + 6. Rewrite: 30 ÷ 5 × 2 + 4 − 6. Evaluate: 30 ÷ 5 = 6, 6 × 2 = 12, 12 + 4 − 6 = 10.
BODMAS reminders
| Step | Rule | Example |
|---|---|---|
| B | Brackets first | (6 + 2) × 3 = 24 |
| O | "Of" means multiply, done before ÷ and × | 1/2 of 20 ÷ 5 = 10 ÷ 5 = 2 |
| D, M | Division and multiplication, left to right | 48 ÷ 6 × 2 = 16 |
| A, S | Addition and subtraction, left to right | 20 − 5 + 3 = 18 |
Type 2: interchanging signs or numbers
The equation is false as written; one pair of signs (or numbers) must be swapped to make it true. Evaluate the original first, then test each option by rewriting and evaluating.
Worked example: 8 + 3 − 6 ÷ 2 × 5 = 26. As written it gives 8 + 3 − 15 = −4. Swap + and ×: 8 × 3 − 6 ÷ 2 + 5 = 24 − 3 + 5 = 26, which balances. Checking the rest: swapping − and ÷ gives −1.5, + and − gives 20, ÷ and × gives 8.6. Exactly one swap works.
Type 3: choosing the signs
Blanks between numbers must be filled with a sign sequence to make the equation true. Try options in turn. Example: 8 _ 2 _ 3 _ 6 = 18. Option ×, +, −? 16 + 3 − 6 = 13. Option ÷, ×, +? 4 × 3 + 6 = 18 — correct.
Type 4: hidden-rule operations
A made-up symbol links two numbers (3 # 4 = 25). Find a rule that fits every example given, then apply it. Test these in order:
| Rule | Check with 3 and 4 | Result |
|---|---|---|
| a² + b² | 9 + 16 | 25 |
| (a + b)² | 7² | 49 |
| a² − b² (or reverse) | 16 − 9 | 7 |
| a × b + (a + b) | 12 + 7 | 19 |
| a × (a + b) | 3 × 7 | 21 |
| a³ + b | 27 + 4 | 31 |
Worked example: if 5 & 2 = 21 and 6 & 3 = 27, what is 8 & 4? Test a² − b²: 25 − 4 = 21, 36 − 9 = 27. Both fit, so 8 & 4 = 64 − 16 = 48.
Type 5: number to number rules
Pairs like 3 → 11, 5 → 27 are hidden-rule questions with one input: here n² + 2 fits (9 + 2, 25 + 2), so 7 → 51. Check squares, cubes, n × (n + 1) and multiples before anything exotic.
Test yourself: 10 questions, 60 s each
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