From Newsgroup: rec.puzzles
On Sun, 28 Dec 2025 19:04:43 -0000 (UTC), David Entwistle wrote:
From '536 Puzzles and Curious Problems' by Henry Ernest Dudeney.
The sides and height of a triangle are four consecutive whole numbers.
What is the area of the triangle?
Dudeney provides some interesting information along with his solution. His solution agrees with Richard Harnden's original solution.
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*230 FIND THE TRIANGLE*
The sides of the triangle are 13, 14, 15, making 14 the base, the height
12 and the area 84. There is an infinite number of rational triangles
composed of three consecutive numbers, like 3, 4, 5, and 13, 14 and 15,
but there is no other case in which the height will comply with our conditions.
The triangles having three consecutive numbers for their sides, and
having an integral area, are:
3, 4, 5
13, 14, 15
51, 52, 53
193, 194, 195
723, 724, 725
etc.
They are found very simply:
52 = 4 x 14 - 4
194 = 4 x 52 - 14
724 = 4 x 194 - 52
or generally, U(subscript n) = 4 x U(subscript n - 1) - U(subscript n -
2), or the general mathematical formula: Find x, so that 3(x^2 - 1) = a perfect square, where 2x, 2x + 1, 2x - 1 are the sides of the triangle.
--
David Entwistle
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