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Consecutive 8-Digit Heron (Posted on 2009-11-15) Difficulty: 3 of 5
The side lengths of a Heronian Triangle are R-1, R and R+1, where R is an 8-digit positive integer which does not contain any leading zero.

Determine all possible value(s) of R for which this is possible.

No Solution Yet Submitted by K Sengupta    
Rating: 3.0000 (1 votes)

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Solution Solution Comment 4 of 4 |

Possibly the hardest part of this puzzle is finding a vaguely geometrical connection to what looks mainly like algebra.

The area formula for an equilateral triangle is A = 3^(1/2)/4*a^2, where a is the length of a side.

Heron's Area Formula is A = (s(s-a)(s-b)(s-c))^(1/2), where s = 1/2(a+b+c), the sum of the three sides.

Let a=(x-1), b=x, c=(x+1); clearly s = 3x/2. Now we substitute for a,b,c:

A = ((3x/2)((3x/2)-(x-1))((3x/2)-x)((3x/2)-(x+1)))^(1/2)  =3^(1/2)/4*((x^2)* (x^2 - 4))^(1/2). The initial term makes the geometrical connection with equilateral triangles tolerably clear.

As mentioned by others, if we want integer x, we must obviously select it so that (x^2 - 4))^(1/2) includes an integer multiple of 3^(1/2), while otherwise being a square, i.e. x^2 - 4 = 3y^2, for some y.

This leads very quickly to the recurrence x = (2 + sqrt(3))^n + (2 - sqrt(3))^n, where x= {4,14,52,194,...}.

(Parenthetically, the Heron Area is then most easily defined in terms of these x's, e.g. A = x/4*(3*(x-2)(x+2))^(1/2), which Excel copes with very easily.)

Now it is just a question of running through the recurrence until we find a qualifying solution: when n=13, x=27246964, the only solution for R compliant with the terms of the problem. This triangle has an area of 321467351292366; the corresponding equilateral triangle has an area of 321467351292366.866..., very close indeed to the integer solution.

Incidentally (2 + (3)^(1/2))^2 = 7+4*3^(1/2), a number relevant to today's puzzle.

Edited on December 18, 2016, 7:10 am
  Posted by broll on 2016-12-18 07:00:32

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