 All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars  perplexus dot info  Squaring the triangle (Posted on 2013-01-07) 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.) 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:

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