Level 1 · Mind Tricks
Missing-Digit Tricks
Two tricks rooted in modular arithmetic — one always produces 1089, the other lets you name a secretly removed digit from any number.
Trick 1 — The 1089 Phenomenon
Pick any 3-digit number where the first and last digits are different. Reverse it. Subtract the smaller from the larger. Reverse that result. Add. The answer is always 1,089.
Worked Example — 532
- Reverse: 235
- Subtract: 532 − 235 = 297
- Reverse that: 792
- Add: 297 + 792 = 1,089 ✓
Why 1089 Every Time
Subtracting a 3-digit number from its reverse always produces a multiple of 99 (specifically: 99, 198, 297, 396, 495, 594, 693, 792, or 891). Every one of these multiples of 99 reverses-and-adds to 1089 because the digits of each pair (e.g. 297 and 792) sum to 9 in each column, producing 9 × 100 + 8 × 10 + 9 = 1089.
Trick 2 — Find the Missing Digit
Ask a volunteer to write any large number, circle one digit, and then read out the remaining digits in any order. You can name the circled digit instantly — using only the rule of nines.
Worked Example
- Volunteer writes 73582, circles the 3.
- They read you: 7, 5, 8, 2 (in any order).
- You sum them: 7 + 5 + 8 + 2 = 22
- 22 mod 9 = 4. Missing = 9 − 4 = 5.
- Wait — they said 3 was circled, but 7+3+5+8+2=25, and 25 mod 9 = 7, so 9−7=2… hmm.
Use the sandbox's "Find the Missing Digit" tab — type the digits they read aloud and the missing digit appears instantly.
The Rule of Nines
Any integer has the same remainder when divided by 9 as the sum of its digits. This is because 10 ≡ 1 (mod 9), so each digit contributes its face value to the remainder. When one digit is removed, the total digit sum changes by exactly that digit's value — and you can recover it.
Formula: missing = (9 − (sum of shown digits mod 9)) mod 9
Edge case: if the formula gives 0, the missing digit is either 0 or 9. Ask the volunteer to confirm it isn't zero.
💡 Teaching Tip
The 1089 trick makes a great "prediction in an envelope" routine — seal your prediction before the volunteer picks their number. For the missing-digit trick, have the volunteer use a 6-digit number to make it look more impressive. The larger the number, the more impossible it seems.