Calculating the volume of our triangular pyramid from last week is quite simple: remember that the volume of a pyramid is one-third the base area times the height, even though the “official” base area is the largest triangle (the one divided by the three hypotenuses) we can take any of the surfaces as a base, for example the right-angled triangle with legs a and b; the third leg, c, is the height and therefore the volume is V = ab/2 xc/3 = abc/6. From here it is easy, although cumbersome, to calculate the height h corresponding to the largest base: calling S the area of that base, we must:
V = Sh/3 = abc/6, from which h = abc/2S
We can calculate S from the three sides of the triangle formed by the three hypotenuses: √(a2+b2), √(a2+c2), √(b2+c2) by applying Heron's formula:
S = √[s(s-a)(s-b)(s-c)] where s is the semi-perimeter of the triangle: (a+b+c)/2
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This is the conceptually simplest way to find h; But see last week's comments section for more subtle suggestions from our smart regular commenters.
And since abc is the volume of a cuboid with edges a, b and c, the following geometric puzzle can be posed: Using six trirectangular tetrahedrons with legs 1, 2 and 3, form a 1x2x3 orthohedron (and/or another compact Body made of same volume as 6 cubic units).
A mental asylum
And speaking of our usual astute commentators, they haven't paid any attention to the Queen and Tiger puzzles, perhaps considering them too simple, so I'll insist on another of the same kind that's a little more complicated (as well ). scary).
The method of Dr. Brea and Professor Pluma is not one of Edgar Allan Poe's best-known stories, but it is one of the most disturbing. And based on this, Raymond Smullyan wrote a no less disturbing riddle:
Inspector Craig from Scotland Yard attended one of Dr. Brea and Professor Pluma, which housed a number of patients as well as other doctors. A resident was called “special” if he believed he was a patient and “special” if all patients believed he was special but no doctor. Craig determined that at least one resident was healthy and that the following condition was met: Each resident had a close friend, and in a pair of residents, A and B, if A believed that B was special, the close friend of A, that B is something special, special. B was a patient. Following this discovery, Craig met privately with Dr. Brea and Professor Pluma. Here's what he spoke to the first one:
-Tell me, Dr. Brea, are all the doctors in this institution healthy?
-But of course!
-And the patients, are they all crazy?
– At least one is.
Next, Craig had the following dialogue with Professor Pluma:
-Dr. Brea says at least one of the patients is crazy, is that true?
-Of course it's true, all patients are crazy!
-And the doctors, are they all healthy?
– At least one is.
-Is Dr. Brea sane?
-But of course! How dare you ask me such a thing?
What conclusion did Inspector Craig come to when he knew that all the beliefs of sensible people are right and all the beliefs of mad people are wrong, and that both are completely sincere and always say what they believe?
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