Gladly — this one’s as old as the hills. The difference between any two numbers that are anagrams of one another is a multiple of 9, which means that the digits add up to 9 (or 18, or 27…) So your “missing” number is whatever digit is required to bump the sum up to the next multiple of 9. Hence the ban on 0 — if you typed in, say, 45, the missing digit could otherwise be either 0 or 9.
The difference between any two numbers that are anagrams of one another is a multiple of 9
To expand on this: suppose that a digit d appears in the numbers. Count the digits, right to left, starting with 0 (i.e. the ones place is 0, the tens place is 1, …). The digit d appears in position a in the first and position b in the second. The two numbers are then:
(… + d*10a + …)
(… + d*10b + …)
Because addition and subtraction are commutative, when you subtract one from the other, you can pair up the terms by digit, so that you’ll have a series of things like:
… + (d*10a – d*10b) + …
Factor out the d, and–supposing that a is greater than b–factor out 10b, and you get:
… + d*10b*(10a-b-1) + …
Since 10 to any power minus 1 is a series of nines, it’ll be divisible by 9. That means each term in the sum is divisible by 9, which means the whole thing is divisible by 9. QED
Darnit, in spite of the promise that “you may use HTML tags for style”, the comments ate my handy superscript markers. Superscript “a”, “b”, and “a-b” throughout.
Gladly — this one’s as old as the hills. The difference between any two numbers that are anagrams of one another is a multiple of 9, which means that the digits add up to 9 (or 18, or 27…) So your “missing” number is whatever digit is required to bump the sum up to the next multiple of 9. Hence the ban on 0 — if you typed in, say, 45, the missing digit could otherwise be either 0 or 9.
The difference between any two numbers that are anagrams of one another is a multiple of 9
To expand on this: suppose that a digit d appears in the numbers. Count the digits, right to left, starting with 0 (i.e. the ones place is 0, the tens place is 1, …). The digit d appears in position a in the first and position b in the second. The two numbers are then:
(… + d*10a + …)
(… + d*10b + …)
Because addition and subtraction are commutative, when you subtract one from the other, you can pair up the terms by digit, so that you’ll have a series of things like:
… + (d*10a – d*10b) + …
Factor out the d, and–supposing that a is greater than b–factor out 10b, and you get:
… + d*10b*(10a-b-1) + …
Since 10 to any power minus 1 is a series of nines, it’ll be divisible by 9. That means each term in the sum is divisible by 9, which means the whole thing is divisible by 9. QED
Darnit, in spite of the promise that “you may use HTML tags for style”, the comments ate my handy superscript markers. Superscript “a”, “b”, and “a-b” throughout.