The ‘hitchhiker triangle’ and the problem of perimeter = area

Abstract

In a recent Student's problem in the Gazette, a triangle is formed from three rectangular strips of paper lying on a flat surface. Their widths were given and the challenge was to find the sides of the triangle so formed. These turned out to be in the proportions 7: 15: 20 [1].The triangle with sides , , is a curious one. It has both perimeter and area ( ) equal to 42. In his Hitchhiker's Guide to the Galaxy Douglas Adams thought '42' the 'answer to the ultimate question of life, the universe, and everything' and it has achieved cult status. For this reason, we may term this particular triangle the 'hitchhiker triangle'.A B C a b c 42 FIGURE 1: The hitchhiker triangle The investigation of integer-sided triangles with perimeter = area originated in the nineteenth century with mentions in the Lady's Diary (1828) and the Lady's and Gentleman's Diary (1865) [2]. In 1904 William Allen Whitworth (1840-1905) and Daniel Biddle (1840-1924) added to triangle lore by proving the startling result that there are exactly five such triangles with this property, one of which is the hitchhiker triangle.Whitworth set the question in the Educational Times, and provided an elegant solution [3]. Biddle also proved the result and made the interesting aside that such triangles could be regarded as the subtraction of one rightangled triangle from another [4].Ensuing problems of this kind are accessible at school level and offer an avenue into geometry, number theory, algorithms, and computer programming. They also present theoretical challenges. We consider an extension of the 'perimeter = area' property and offer a proof of a conjecture on the longest possible side of these integer-sided triangles [5].

Less than equable Heronian triangles

“…We shall use a standard method for considering integer-sided triangles, based upon the incircle, [2].…”

“…A proof of this result was given by Whitworth [1]. Since Whitworth's time, much attention has been given to triangles whose areas are integer multiples of their perimeters, for example [2,3]. However, as this paper will show, Heronian triangles with areas less than their perimeters have some mathematical interest.…”

“…x a a x = 2X a = 6A X (X + 2) = 3A 2 The greatest common divisor of and is either 1 or 2 and so there are positive integers and such that…”

100.16 The Diophantine equation n(x + y + z) = xyz

“…In [1], an investigation of integer-sided triangles led to a number of interesting results, including a proof of a proposition equivalent to the result given below. In this Note, the repeated application of an elementary inequality will be used to obtain a simple direct proof.…”

Diophantine Equations Relating Sums and Products of Positive Integers: Computation-Aided Study of Parametric Solutions, Bounds, and Distinct-Term Solutions

“…Upper bounds for the solution terms or their sum are needed for computational enumeration. In 2015, Tony Crilly and Colin Fletcher found that the maximum semiperimeter of an integer-sided triangle tightly wrapping a circle of a radius r is (r 2 + 1)(r 2 + 2) [23]. The semiperimeter in their calculation equaled the sum of solution terms of the Diophantine Equation ( 8) with F = r 2 ; thus, they established an upper bound for the sum of solution terms of (8) to be (F + 1)(F + 2).…”