Solution to: Missing Pages

Let the number of missing pages be n and the first missing page p + 1. Then the pages p + 1 up to and including p + n are missing, and n times the average of the numbers of the missing pages must be equal to 9808:

n × (((p + 1) + (p + n)) / 2) = 9808

In other words:

n × (2p + n + 1) / 2 = 2 × 2 × 2 × 2 × 613

So:

n × (2p + n + 1) = 2 × 2 × 2 × 2 × 2 × 613

One of the two terms n and 2p + n + 1 must be even, and the other one must be odd. Moreover, the term n must be smaller than the term 2p + n + 1. It follows that there are only two solutions:

• n = 1 and 2p + n + 1 = 2 × 2 × 2 × 2 × 2 × 613, so n = 1 and p = 9808, so only page 9808 is missing.
• n = 2 × 2 × 2 × 2 × 2 and 2p + n + 1 = 613, so n = 32 and p = 290, so the pages 291 up to and including 322 are missing.

Because it is asked which pages (plural) are missing, the solution is: the pages 291 up to and including 322 are missing.

Back to the puzzle