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Question

Alex added the page numbers of a book together and got a total of 888. Unfortunately, he didn’t notice that one of the sheets of the book was missing with an odd page number on the front and an even page number on the back. What was the page number on the final page in the book?

Answer 1

You can solve this problem by guessing and checking, but there are a few useful math processes that will save us some time.

 

 

First, think about adding all the numbers between 1 and 10 together. You get:

 

1+2+3+4+5+6+7+8+9+10 = 55

 

This sum (55) can be calculated quickly in your head by pairing up the numbers like so:

 

(1+10)+(2+9)+(3+8)+(4+7)+(5+6) = 11+11+11+11+11 = 11×5 = 55

 

Each pair of numbers is equal to the low number (1) plus the high number (10), and since 10/2 = 5, there are 5 pairs of numbers. So, we can do the summation like this:

 

(10+1)×(10/2) = 11×5 = 55

 

That shortcut helps add any chunk of consecutive numbers quickly. 1 through 100, for example, is (100+1)×(100/2) = 101×50 = 5050.

 

That means, given the number of pages in a book, we can quickly find out what the page numbers' sum is.

 

 

It's useful to be able to calculate that sum, but that's only half the battle. Alex didn't notice that his book was missing a page, so he didn't get exactly the answer we would from summing all the numbers. That doesn't mean his count is totally useless, though. Here are a few things we'll look for when guessing and checking:

 

1. We want our sum to be greater than 888 (Alex's book was missing two pages, subtracting from the sum)

 

2. We want our sum to be not much greater than 888 (Alex's book was only missing two pages, and even if it was the last two pages, he could only be missing up to 2×(number of total pages) from his count)

 

3. We want our sum to be greater than Alex's by an odd number (Alex's book was missing an odd page and an even page; since an odd number plus an even number is an odd number, his count is off by some unknown odd number)

 

 

Now we can start guessing and checking. From earlier, we know that adding up the pages in a 100-page book gives us a sum of (100+1)(100/2) = 5050, which would be way too high. So let's guess 50:

 

(50+1)(50/2) = (51)(25) = 1275

 

Still too high. It fails criterion (2). Let's try 40:

 

(40+1)(40/2) = 820

 

That's much closer, but now we're too low. It fails criterion (1). 41 would be 820 + 41 = 861, which is still too low. So let's try 42:

 

861 + 42 = 903

 

That fits criteria (1) and (2), being not too low or too high. It's also greater than 888 by an odd number: 15.

 

That means Alex could have a 42-page book that's missing pages 7 and 8.

 

 

Let's try 43 and 44 just to make sure there isn't another number that would fit:

 

903 + 43 = 946

 

That's close, but greater than 888 by 58, an even number. It fails criterion (3).

 

946 + 44 = 990

 

That's too high. It fails criterion (2). So, the right answer is indeed 42.