Integer Triangles (1)

Integer Triangle is an area of Geometry and Number theory in which a lot of work has been added in the past decade. A lot of questions on Integer triangles are part of Recreational Mathematics now. Besides many examiners have started including such questions in different aptitude tests.

An integer triangle is a triangle whose all sides have lengths that are integers.

There are a few general properties for an integer triangle. 

Finding no. of Integer Triangles when Perimeter is given:

Suppose we have to find the no. of Integer Triangles when the perimeter is given as equal to 20.

Here we use the triangular inequality and realize that the longest side of such triangle can be 9 and we can tabulate the data as below:

N0.	Longest Side a	Side b	Side c	Perimeter
1	9	9	2	20
2	9	8	3	20
3	9	7	4	20
4	9	6	5	20
5	8	8	4	20
6	8	7	5	20
7	8	6	6	20
8	7	7	6	20

Now it can be seen that calculating in this manner is very cumbersome. If the perimeter given is higher it will take much longer time.

However we can use a simple formula for such calculation.

Now there may be two cases; the perimeter may be even or odd

Case 1:  when the perimeter, say p, is even:

The no. of triangles is given by the formula nint (p²/48)

Where, nint means nearest Integer Function.

For example, nint (1.6) is equal to 2 and nint (3.2) =3.

( Additionally half-integers are always rounded to even numbers For example, nint(1.5)=2, nint(2.5)=2, nint(3.5)=4, nint(4.5)=4, etc.)

For our example of perimeter equal to 20; we can calculate in this manner

nint (20x20/48)= nint (8.33) =8

Case 2: when the perimeter, say p, is odd:

The no. of triangles is given by the formula nint {(p+3)²/48}

Finding no. of Integer Triangles when longest side is given:

Case 1:  When the longest, say a, is even.

No. of Integer triangles = a(a+1)

Case 2:  When the longest, say a, is odd.

No. of Integer triangles = a²

 ( To be Continued)

Maths Books

There are many books on Recreational Mathematics and Pre College Maths which even laypersons can appreciate and enjoy.  These books may be helpful in preparing for Quantitative Aptitude also as many questions are adapted by the examiners from these books. Given below is a list of good books, in random order, one may wish to read at leisure.

(1)   Golden Ratio by Mario Livio

(2)   Fermat’s Last Theorem by Simon Singh

(3)   Story of e by Eli Maor

(4)   The Magic of Numbers by Eric Temple Bell

(5)   The Lighter side of Mathematics by Richard K Guy

(6)   The Colors of Infinity by Prof Ian Stewart

(7)   Cabinet of Mathematical Curiosities by Ian Stewart

(8)   Mathematical Recreations by Martin Gardener

(9)   The Little book of Big Primes by Paulo Ribenboim

(10)  Math Charmers by Posamantier

(11)  Puzzles to Puzzle you by Shakuntala Devi

(12) Joy of Mathematics by Theonny Pappas

(13)  The Penguin Dictionary of Curious and Interesting Numbers by David Wells

(14) 123…Infinity by George Gamov

(15) Amusements in Mathematics by H E Dudeney

 (16) Pi Unleashed by Jorg Arndt

(17) The pleasures of Pi, e and other interesting Numbers by Y E O Adrian

(18) The Fabulous Fibonacci Numbers by Posamantier

(19) The simple solution to Rubik's Cube by Nourse

(20) Dr. Euler's fabulous Formula by Paul J Nahin

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Perfect Triangle

Perfect Triangles are defined as the triangles having side lengths that are integers and for which the area and perimeter of the triangle are equal.

Imagine a triangle with side lengths of x + y, y + z, and x + z, the problem can be narrowed down algebraically. For this triangle, the perimeter  is equal to 2 (x +y+ z)

 And the semi perimeter ‘s’ =  (x +y+ z) 

Equating area with the semi perimeter we get:

√[s{s-(x +y)}{s-(y +z )} {s-( z+x)}] = 2 (x +y+ z)

√ [(x +y+ z){(x +y+ z) -(x +y)}{(x +y+ z) -(y+z )}{ (x +y+ z) -(z+x)}]= 2(x +y+ z)

Now solving the above equation , we get the following equation:

xyz = 4(x+y+z)

We have already solved the above equation  in the previous post on Diophantine Equation:

The solutions to the above equation are as below:

x	y	z	x+y Side 1 of the Triangle	y+z Side 2 of the Triangle	z+x Side 3 of the Triangle
24	5	1	29	6	25
14	6	1	20	7	15
9	8	1	17	9	10
10	3	2	13	5	12
6	4	2	10	6	8

Thus there are only five Perfect Triangles for which The Perimeter and the area are same numerically as given in the above table.