From Newsgroup: rec.puzzles

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?
--
David Entwistle
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From Newsgroup: rec.puzzles

On 28/12/2025 19:04, 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?

Using Heron's Fomular <https://en.wikipedia.org/wiki/Heron%27s_formula>

The height (h) must be less than the shortest side (a),
so: h < a (=h+1) < b (=h+2) < c (=h+3)

The area (A) is 0.25 * sqrt((4a^2b^2) - (a^2 + b^2 - c^2)^2
And check the height with: 2A / b

Wrote a program to brute force that and the only thing that works is:

$ ./a.out
h = 12, a = 13, b = 14, c = 15 ... area = 84









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From Newsgroup: rec.puzzles

In article <10irv0b$f5oh$1@dont-email.me>,
David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> 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?

0

The sides are 1, 2, and 3.

-- Richard


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From Newsgroup: rec.puzzles

In article <10j0b6e$1nsg6$1@dont-email.me>,
Richard Harnden <nospam.harnden@invalid.com> wrote:

The height (h) must be less than the shortest side (a),

Not true of triangles in general - for example, a 3,4,5 triangle has
height 4 from the 3 side.

At least two of the sides are at least as long as the height, but we
could conceivably have a triangle where a, h, b, c are consecutive
integers. Suppose there is one.

Writing h=a+1, b=a+2, c=a+3 we can use Heron's formula to get

4a^2(a+1)^2 = (3a+5)(a+5)(a+1)(a-1)

which is a quartic. Solving numerically (google will do this for you!)
we get a ~= 1.262 so the triangle with sides 1.262, 3.262, 4.262
has a height 2.262. Unfortunately these are not integers.

We can use the same approach to find the correct answer. If the
height is smaller than any of the sides, we have a=h+1, b=h+2, c=h+3
and depending on which side we take as the base we have

4h^2(h+1)^2 = (3h+6)(h+4)(h+2)h
or
4h^2(h+2)^2 = (3h+6)(h+4)(h+2)h
or
4h^2(h+3)^2 = (3h+6)(h+4)(h+2)h

The first and third need numerical solution and produce non-integers.
But the middle one gives us

4h(h+2) = (3h+6)(h+4)
4h = 3(h+4)
h = 12

-- Richard
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From Newsgroup: rec.puzzles

On Wed, 31 Dec 2025 12:49:14 -0000 (UTC), richard@cogsci.ed.ac.uk
(Richard Tobin) wrote:

In article <10j0b6e$1nsg6$1@dont-email.me>,
Richard Harnden <nospam.harnden@invalid.com> wrote:

The height (h) must be less than the shortest side (a),

This is a remarkable insight, but I had never heard it said
before .....

Not true of triangles in general - for example, a 3,4,5 triangle has
height 4 from the 3 side.

... perhaps because it is not true in general. Richard has already
given an example, but this could be "worse". When you consider
obtuse angle triangles, you can make one of the sides including
the obtuse angle as small as one likes, while maintaing the same
altitude. This is well illustrated in Wikipedia's entry for triangle
altitudes:

https://en.wikipedia.org/wiki/Altitude_(triangle)

In the first figure, we can make BC as small as we wish while
keeping the altituted AD fixed.

We can use the same approach to find the correct answer. If the
height is smaller than any of the sides, we have a=h+1, b=h+2, c=h+3
and depending on which side we take as the base we have

4h^2(h+1)^2 = (3h+6)(h+4)(h+2)h
or
4h^2(h+2)^2 = (3h+6)(h+4)(h+2)h
or
4h^2(h+3)^2 = (3h+6)(h+4)(h+2)h

The first and third need numerical solution and produce non-integers.
But the middle one gives us

4h(h+2) = (3h+6)(h+4)
4h = 3(h+4)
h = 12

-- Richard

Excellent!


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From Newsgroup: rec.puzzles

On Wed, 31 Dec 2025 11:36:27 -0000 (UTC), richard@cogsci.ed.ac.uk
(Richard Tobin) wrote:

In article <10irv0b$f5oh$1@dont-email.me>,
David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> 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?

0

The sides are 1, 2, and 3.

-- Richard

Even better! You need a special medal for this one!!
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From Newsgroup: rec.puzzles

On Wed, 31 Dec 2025 16:26:21 -0500, Charlie Roberts
<croberts@gmail.com> wrote:

On Wed, 31 Dec 2025 11:36:27 -0000 (UTC), richard@cogsci.ed.ac.uk
(Richard Tobin) wrote:

In article <10irv0b$f5oh$1@dont-email.me>,
David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> 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?

0

The sides are 1, 2, and 3.

-- Richard

Even better! You need a special medal for this one!!

Actually, I raced past this without thinking. In his post,
Richard H states that

The height (h) must be less than the shortest side (a),
so: h < a (=h+1) < b (=h+2) < c (=h+3)
<<

Fleetingly, I thought of h = 0 (a valid integer) .... but
I was not bright enough to catch the irony and subtlety :-(


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From Newsgroup: rec.puzzles

richard@cogsci.ed.ac.uk (Richard Tobin) writes:

In article <10irv0b$f5oh$1@dont-email.me>,
David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> 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?

0

The sides are 1, 2, and 3.

You degenerate, you!

Phil
--
We are no longer hunters and nomads. No longer awed and frightened, as we have gained some understanding of the world in which we live. As such, we can cast aside childish remnants from the dawn of our civilization.
-- NotSanguine on SoylentNews, after Eugen Weber in /The Western Tradition/
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From Newsgroup: rec.puzzles

On Sun, 11 Jan 2026 15:24:40 +0200, Phil Carmody <pc+usenet@asdf.org>
wrote:

richard@cogsci.ed.ac.uk (Richard Tobin) writes:

In article <10irv0b$f5oh$1@dont-email.me>,
David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> 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?

0

The sides are 1, 2, and 3.

You degenerate, you!

Phil

Right on!

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