## Digit Exchange

A three-digit number increases by nine if you exchange the second and third digits. The same three-digit number increases by $90$ if you exchange the first and second digits. By how much will the number increase if you exchange the first and third digits of the same number?
Source: mathcontest.olemiss.edu 11/14/2011

SOLUTION
Exchange second and third digits
Suppose the three-digit number is $a49$. If we exchange the second and third digits, the resulting number is $a94$. The increase $a94-a49=45$ is much greater than 9.
So we try a smaller number for example $a45$ and exchange the second and third digits to obtain $a54$. The increase equals $a54-a45=9$ which is the correct amount. But, this is true also of $a23, a34$, etc.

Exchange first and second digits
From what we learned above the first digit should be 1 less than the second digit if we want the increase to equal $90$. For example, if we start with the three-digit number $123$ and exchange the first and second digits to obtain $213$, the increase equals $213-123=90$ as expected. But, this is true of the many more numbers as shown below.

 123 213 213 – 123 = 90 234 324 324 – 234 = 90 345 435 435 – 345 = 90 456 546 546 – 456 = 90 567 657 657 – 567 = 90 678 768 768 – 678 = 90 789 879 879 – 789 = 90

Exchange first and third digits
If we exchange the first and third digits, the increase equals 198.

 123 321 321 – 123 = 198 234 432 432 – 234 = 198 345 543 543 – 345 = 198 456 654 654 – 456 = 198 567 765 765 – 567 = 198 678 876 876 – 678 = 198 789 987 987 – 789 = 198