Home > Problem of the Week > Archive List > Detail

<< Prev 4/16/2006 Next >>

The Puzzle Book

Wayne and Reg wrote:

We have this puzzle book and one day noticed that there was a page missing. We then found the most amazing thing about our book, the sum of the numbers on the remaining pages was 10,000. Incredible! The questions are: Which page is missing and how many pages were in the book?

A few days later Dennis called Reg, he said, "Hello" and heard "You fool, thereís more than one answer to the book problem." Click.

Reg wonders who that could have been and what was going on. He works on the problem a bit and figures it was Dennis that called. So, he calls Dennis and says, "Youíre the fool; they donít print books like that."

Which page was missing?
How many pages in the box?
What is going on regarding the last comment?

 

Source: Reg Waddoups


Hint: If a page is missing, the page numbers x-1 and x are actually missing. The remaining page numbers are (x+1), (x+2), ...., (x+z).

Now, play with the sum (x+1)+(x+2)+....+(x+z), which must equal 10,000. How can you rewrite the sum as a quadratic?

Hint: Taking historical liberties, think about Gauss' clever process for summing sequential whole numbers!

 


Solution Commentary: Using the hint, you should get these equations:
(x+1)+(x+2)+....+(x+z) = 10000
zx+(z)(z+1)/2 = 10000
z2+(2x+1)(z)-20000 = 0
Now, it is up to you...use the quadratic equation with x as a parameter (natural number). You also know that z has to be a whole number.

After some playing, you should discover that x and z can have multiple values:
x = 17, z = 125 implies page numbered 16-17 is missing, and the book has 142 pages
x = 296, z = 32 implies page numbered 295-296 is missing, and the book has 328 pages
x = 387, z = 25 implies page numbered 386-387 is missing, and the book has 412 pages

So what is the problem with having the three solutions? Look at the pages of a book and focus on how they are numbered.....