A three-digit number increases by nine if you exchange the second and third digits. The same three-digit number increases by 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 . If we exchange the second and third digits, the resulting number is . The increase is much greater than 9.

So we try a smaller number for example and exchange the second and third digits to obtain . The increase equals which is the correct amount. But, this is true also of , 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 . For example, if we start with the three-digit number and exchange the first and second digits to obtain , the increase equals 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 |

**Answer**: 198.