Why the Reversed-Number Trick Always Lands on a Multiple of Nine
In a math circle, students are investigating what happens when a two-digit number is written backwards and subtracted from the original, larger number first. Four results go on the board: 52 − 25 = 27; 74 − 47 = 27; 81 − 18 = 63; 30 − 03 = 27.
Jonah concludes that the answer is always 27. Maya says 81 − 18 disproves that and proposes the real rule is "the answer is always a multiple of 9." Ravi points out that the tens digit of the answer changes too, so collecting more examples will not settle the question; he wants an argument that covers every two-digit number at once, including numbers ending in 0.
The teacher adds a constraint: the argument must explain why the quotient (the answer divided by 9) sometimes changes and exactly what controls it, and it must predict 91 − 19 before anyone computes it.
1. Write the general argument for two-digit numbers, using a tens digit and a ones digit, and state the rule for the answer and for the quotient.
Read the worked answer
Let the two-digit number be 10a + b, where a is the tens digit (1 through 9) and b is the ones digit (0 through 9). Its reversal is 10b + a.
Subtracting the smaller from the larger is easiest to handle by taking the difference in a fixed direction and then allowing for sign:
(10a + b) − (10b + a) = 9a − 9b = 9(a − b).
So the difference is always 9 times the gap between the digits. If a > b the difference is positive; if a < b it is negative and we take the absolute value, which is why the worked examples all write the larger number first.
Check against the board: 52 − 25 gives a − b = 3, so 9 × 3 = 27 ✓. 74 − 47 gives a − b = 3 ✓. 81 − 18 gives a − b = 7, so 9 × 7 = 63 ✓. 30 − 03 gives a − b = 3 ✓.
The quotient is not constant; it is exactly the distance between the two digits. That is the quantity the class was actually watching change.
2. Use the argument to adjudicate between Jonah and Maya, and predict 91 − 19 without computing it directly from the subtraction.
Read the worked answer
Jonah's rule is a coincidence of three examples that happened to use digits three apart. Four out of the five data points the class had (52/25, 74/47, 30/03) share a digit gap of 3, which is why 27 kept appearing; 81 and 18 have a gap of 7 and break the pattern. A rule that three of four examples support is not thereby the rule.
Maya's rule is correct but incomplete: every difference is a multiple of 9, but she has not said which multiple, so her rule cannot predict anything beyond divisibility. The algebraic version is strictly stronger - it entails her rule and pins the value.
For 91 − 19: the digit gap is 9 − 1 = 8, so the difference must be 9 × 8 = 72. Direct subtraction confirms it. The prediction came from the structure, not from the example list, which is exactly what Ravi asked for.
3. Extend the argument to three-digit numbers and explain why the middle digit plays no role in the result.
Read the worked answer
Write the three-digit number as 100a + 10b + c, with a nonzero. Its reversal is 100c + 10b + a.
Subtracting: (100a + 10b + c) − (100c + 10b + a) = 99a − 99c = 99(a − c).
The middle digit cancels because it contributes exactly 10b in both numbers - moving a digit to the mirrored position leaves the middle term untouched. So every three-digit reversal difference is a multiple of 99, and the quotient is the gap between the outer digits.
Check: 731 − 137 = 594, and 99 × (7 − 1) = 594 ✓. As with two digits, the sign depends on which outer digit is larger, so the absolute difference is 99|a − c|.
The pattern generalizes: for an n-digit number, the reversal difference is divisible by 9 and by 11 whenever the digit count is even, and the size is governed entirely by the outer digits. That is why the trick feels magical and is really just place value.