You may have noticed that each book published in recent years has an International Standard Book Number (ISBN).

For example, I have a text book with the number ISBN 0 412 09810 5. The first digit (0) indicates that the book is written in English. The second string (412) indicates the publisher. The third string (09810) is a title number. And the tenth digit (5) is a check digit used to guard against transcription errors and misreading. We might note that the check ‘digit’ is sometimes 10 (written X).

Now there is a simple rule which relates the check digit to the other nine digits. To determine this rule you will need to make a small collection of ISBN numbers. Numbers close together will be helpful; for example: ISBN 0 04 793045 4, ISBN 0 04 793046 2.

Now, what is the check digit for this book taken from my shelf: ISBN 0 201 10405 _ ?

Hint 1

Make a small collection of numbers as suggested.

Hint 2

Any rule will need to be simple, and will need to act differently on each digit –– for example it must distinguish a simple interchange of two digits.

                   

Solution

Making a small collection of ISBN numbers is probably most helpful for checking a rule rather than for deriving it. Suppose that the ten digits in the ISBN number are d₁, d₂, d₃, ..., d₉, d₁₀ (d₁₀ may be 10). What we are looking for is some rule, or combination, of these numbers which will give a different answer if one digit is altered, or if two digits are accidentally interchanged.

This might suggest that we look at

    1d₁ + 2d₂ + 3d₃ + ... + 9d₉ + 10d₁₀,
or                                                             (*)
    10d₁ + 9d₂ + 8d₃ + ... + 2d₉ + 1d₁₀.

It is obviously no use having a coefficient 0, as any change in that term would remain undetected. Of course the coefficients 1, 2, ... , 10 might occur in any order, but we are looking for a simple rule.

Now suppose we form one or other of the above sums. How will the check digit be chosen? The fact that the check ‘digit’ can be 0, 1, 2, ... , 10 gives us the clue. For these are the possible remainders on division by 11.

So we choose one of the expressions (*), and divide by 11. The check digit is chosen to give remainder zero on division by 11. Your ISBN collection will show you that the second expression in (*) is the one that is used.

For the number ISBN 0 201 10405 _, we obtain

(10 x 0) + (9 x 2) + (8 x 0) + ... + (3 x 0) + (2 x 5) + (1 x d₁₀) = 0.

This gives  57 + d₁₀ to be exactly divisible by 11. Hence d₁₀ = 9.

Extensions

Show that instead of using the coefficients 1, 2, ... , 10,
we could use ±1, ±2, ... , ±5.
 

Hint 1

Hint 2

Solution

Extensions