All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Just Math
Squaring the triangle (Posted on 2013-01-07) Difficulty: 3 of 5

The number 185136 has the interesting property that it is a triangular number that is also the product of 3 consecutive integers.

I square 185136 and each of the x consecutive numbers following it, and total the sum of all these squares, S.

Then I do the same with 185136+x+1 and the y consecutive numbers following it, until I again reach the same sum, S.

Find positive integer values of x,y,S compliant with the above requirements.

See The Solution Submitted by broll    
No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
Hints/Tips re(3): Bit different with added consideration - short step? Comment 7 of 7 |
(In reply to re(2): Bit different with added consideration - short step? by brianjn)

Close enough, Brian.

In fact the range x(=y+1) is always exactly half the index of the triangular number; equivalently: T = 2x^2+x; while S factors nicely as 1/6(x)(x+1)(2x+1)(12x^2+12x+1).

Note that x here is 1 less than the full range: '185136 and each of the x consecutive numbers following it'; and likewise with y: '185136+x+1 and the y consecutive numbers following it', so there is one more square in the first (full) range than there is in the second - in this case, 305 as compared to 304.

A rather neat full proof is easily worked up once the principle is understood.

 

Edited on January 8, 2013, 11:45 am
  Posted by broll on 2013-01-08 07:52:31

Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (3)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information