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CurlyBracket

Member

From: Your Maths Textbook

Registered: 2022-01-03

Posts: 155

Pages of Novels

A leaf is torn from a paper back novel.  The sum of the numbers on the remaining pages is 15000.

What are the page numbers on the torn leaf?

Hint, please!

Last edited by CurlyBracket (2022-05-12 17:07:46)

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Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Pages of Novels

If n is the number of pages in the book and m and m+1 are the torn pages then

As there are two unknowns but only one equation, there's no simple algebraic trick for solving this.  You do know that n and m have to be positive integers, but that's still not much help.

So I looked to see if there was a list of the triangle numbers [n(n+1)/2] on the net but couldn't find one.  So I made my own using a spreadsheet.

I made column A the counting numbers, and column B the triangle numbers using the formula.  Then I had a look.

The first triangle number after 15000 is n = 173, and that comes out to be 15051.  So m = 25 and m+1 = 26 seems to work.  But so does n = 174 m = 112, m+1 = 113  [15225 = 112 - 113]  But that means the first page is even so I'll reject that.

So I made column C = column B - 15000 and column D = column C/2 and then scanned down for a number of the form odd + 0.5

The first I came to was n = 178, n(n+1)/2 = 15931 which makes m = 465 and m=1 = 466.  It fits the arithmetic but you cannot tear out page 465 from a 178 page book. 

After 178 this issue continues because the sum - 15000 excess gets bigger faster than the sum of pages goes up.  So it looks like the first answer I found is the only one.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob