Background for this problem:

When a book is published, it is assigned a number called the International Standard Book Number (ISBN).   From 1970 until 2007, this number consisted of a 10-digit number, but it now consists of a 13-digit number. For example, the ISBN of   UMAP Modules   (1984) is 0-912843-07-1, but the ISBN of Dan Brown's   The lost Symbol   (2009) is 978-0-385-50422-5.

In the 10-digit example above, the first digit, 0, indicates that the book was published in an English-speaking country; the digits 912843 represent the publisher (COMAP, Inc.); 07 are the identifying numbers that COMAP has assigned to the book; the final digit, 1, is called the check digit of the ISBN.
To obtain the ISBN check digit:

1. Multiply the first nine digits of the ISBN by 10, 9, 8, 7, 6, 5, 4, 3, and 2, respectively, and then compute
      the sum of these nine products.

2. Find the remainder when this sum is divided by 11.

3. Subtract the remainder from 11 to determine the check digit.   [NOTE: So that each possible check
      digit is a single digit, a check digit of 10 is written as X and a check digit is assigned the value of 0 if
      there is a remainder of 0.]

For the ISBN 0-912843-07-1 that is discussed above:

1.       10(0) + 9(9) + 8(1) + 7(2) + 6(8) + 5(4) + 4(3) + 3(0) + 2(7) = 197

2.       197 = 11(17) + a remainder of 10.

3. Check digit = 11 - 10 = 1. (It checks!)



Problem of the Month:

There are two parts to this problem:
  1. Determine the check digit for the book whose partial ISBN number is 1-4000-7917-?, where ? is the check digit.

  2. From the experiences of people working with the ISBN, one of the most common errors is the transposition of adjacent digits.   The ISBN 0-45-283527-8 is not correct.   Assume the check digit is correct.   Can you find two adjacent digits that are transposed that would give you a valid ISBN?
Solution to the Problem:

1. The check digit is 9.   The ISBN is 1-4000-7917-9, which is The DaVinci Code written by Dan Brown.

2. There are two possible answers: 0-42-583527-8 and 0-45-283257-8.
        First, note that the check digit for the original nine digits should be 5 instead of 8,
                which is a difference of 3.
        Therefore, you must find 2 adjacent digits whose difference is 3.
                There are two possibilities, each involving a 2 and a 5.

Correctly solved by:

1. James Alarie Flint, Michigan
2. Kayle Rippetoe Mountain View High School,
Mountain View, Wyoming
3. Nathon Taylor Mountain View High School,
Mountain View, Wyoming
4. Jason Stoddard Mountain View High School,
Mountain View, Wyoming
5. Taylor Meeks Mountain View High School,
Mountain View, Wyoming
6. Michael Reilly Silver Spring, Maryland
7. Jitesh Kamboj Bangalore, Karnataka, INDIA