Number Tricks - The Maths Behind the Magic
Number tricks are surprising patterns and shortcuts hidden inside ordinary arithmetic. They look like magic but every single one has a clear mathematical explanation. Learning these tricks develops number sense, makes you faster at calculations, and – once you understand the algebra – deepens your understanding of how numbers truly work.
Trick 1: The 1 089 Mystery
Steps:
1. Choose any 3-digit number where the first and last digits differ by at least 2 (e.g. 731).
2. Reverse the digits (137).
3. Subtract the smaller from the larger (731 − 137 = 594).
4. Reverse the result (495).
5. Add the last two numbers (594 + 495 = 1 089).
The answer is always 1 089.
Why? Algebra shows that (100a + 10b + c) − (100c + 10b + a) = 99(a−c). The possible values of 99(a−c) for a−c ≥ 2 are 198, 297, ..., 891. Reversing and adding each always gives 1 089.
Trick 2: Think of a Number
Steps: Think of a number. Double it. Add 10. Halve the result. Subtract your original number. Your answer is 5.
Why? Let the number be n.
Double: 2n. Add 10: 2n + 10. Halve: n + 5. Subtract n: 5.
The chosen number cancels out completely, leaving only the constant 5.
Trick 3: The Prediction Game
Steps: Think of a number. Multiply by 3. Add 6. Divide by 3. Subtract your original number. Your answer is 2.
Why? n → 3n → 3n+6 → (3n+6)/3 = n+2 → n+2−n = 2.
Once you understand this, you can design your own tricks by controlling the constant that remains.
Trick 4: Divisibility Rules
| Divisor | Rule | Example |
|---|---|---|
| 2 | Last digit is even | 348 ÷ 2 ✓ (8 is even) |
| 3 | Sum of digits divisible by 3 | 471: 4+7+1=12, divisible by 3 ✓ |
| 4 | Last two digits divisible by 4 | 1 732: 32 ÷ 4 = 8 ✓ |
| 5 | Last digit is 0 or 5 | 835 ÷ 5 ✓ |
| 6 | Divisible by both 2 and 3 | 534: even and 5+3+4=12 ✓ |
| 7 | Double last digit, subtract from rest; repeat | 161: 16 − 2 = 14 = 7×2 ✓ |
| 8 | Last three digits divisible by 8 | 4 512: 512 ÷ 8 = 64 ✓ |
| 9 | Sum of digits divisible by 9 | 729: 7+2+9=18, 18÷9=2 ✓ |
| 10 | Last digit is 0 | 5 670 ÷ 10 ✓ |
| 11 | Alternating digit sum divisible by 11 | 5 918: 5−9+1−8 = −11 ✓ |
Trick 5: The 9 Times Table Finger Method
Hold out all 10 fingers. To find 9 × n, fold down finger n (counting from the left). The fingers to the left of the folded finger give the tens digit; the fingers to the right give the units digit.
Example: 9 × 7: fold finger 7. Left = 6 fingers. Right = 3 fingers. Answer: 63.
Why? 9n = 10(n−1) + (10−n) = 10n − 10 + 10 − n = 9n. The pattern holds for n = 1 to 10.
Trick 6: Casting Out Nines (Checking Calculations)
The digit sum of any number, reduced to a single digit, is called its digital root. Digital roots obey the same arithmetic as the numbers themselves (modulo 9).
Use this to check multiplication: if A × B = C, then digital_root(A) × digital_root(B) should equal digital_root(C).
Example: Check 47 × 83 = 3 901.
digital_root(47) = 4+7 = 11 → 2. digital_root(83) = 8+3 = 11 → 2. 2 × 2 = 4.
digital_root(3 901) = 3+9+0+1 = 13 → 4. ✓ Consistent (but 47×83 = 3 901 is actually correct).
Trick 7: Any Number Times 111
For a 3-digit number abc: 111 × abc → write the cumulative sums then write them back down.
Example: 111 × 234:
Leading: 2. 2+3 = 5. 2+3+4 = 9. 3+4 = 7. Trailing: 4.
Answer: 25 974. Verify: 234 × 111 = 234 × 100 + 234 × 11 = 23 400 + 2 574 = 25 974 ✓.
Trick 8: Kaprekar’s Constant
Take any 4-digit number (not all digits the same). Arrange digits largest to smallest, then smallest to largest. Subtract. Repeat. You always reach 6 174 within 7 steps.
Example: 5 432 → 5 432 − 2 345 = 3 087 → 8 730 − 0 378 = 8 352 → 8 532 − 2 358 = 6 174. Done.
Key Takeaways
- Every number trick has an algebraic reason – understanding it means you can create your own.
- Divisibility rules are fast checks that avoid long division.
- Casting out nines lets you verify multiplication results instantly.
- Surprising constants like 1 089 and 6 174 emerge from the structure of our base-10 number system.
Practice Questions
- Apply the 1 089 trick to the number 852. Show all steps.
- Design your own “think of a number” trick that always results in 7. Show the algebra.
- Without dividing, determine whether 4 358 is divisible by 3, 4, 6, 8, and 9.
- Use casting out nines to check whether 68 × 47 = 3 196 is plausible.
- Apply Kaprekar’s routine to 3 141. How many steps does it take to reach 6 174?