Math Q: Calculus, Maximization / Optimization Problem

[Hari_Seldon]

@DPRK

Suppose we somehow define the space of rectangles in the plane: for example, by specifying the center, length, width, and orientation as coordinates. What I care about is that the function which computes the area of a rectangle is a continuous function on this space.

Now consider any triangle; some of the rectangles will be contained in it, which may be a complicated-looking condition on the coordinates but it will define a closed and bounded subset of the space of rectangles. It follows that the area function takes on extreme values.
Well, if an inscribed rectangle has 4 or 3 or 2 of its corners in the interior of the triangle, by slightly enlarging it you can find a rectangle with strictly greater area still contained in the triangle. If a rectangle is jammed in so that only 1 vertex is in the interior, I think that by considering the parallelogram of equal area obtained by sliding the interior vertex parallel to one of the two sides of the rectangle in which it is not contained until it hits the boundary of the triangle, you can see that the resulting parallelogram is contained in a triangle strictly smaller than the original, so the tilted rectangle is again not optimal.

If you can give the space of rectangles a compact topology in which area is a continuous function, your argument for a maximum would be valid, but I doubt you can. If you have access to “What is Mathematics” by Courant and Robbins, read their discussion of the isoperimetric problem. They give an elementary proof that if there is a figure of maximum area with a given perimeter, then it must be a circle, but they wave their hands on the existence of a maximum.

And yes, it seems certain that the maximum rectangle must have its vertices on the triangle, but that still doesn’t prove that a maximum exists.

[DPRK]

But a triangle has 3 sides, so if we place four vertices on its boundary, at least two of them will be on the same edge of the triangle. The segment joining them will be an edge of the rectangle, which you proved leads to half the area of the triangle.

The space of rectangles business was meant to suggest a proof that there is a maximal-area rectangle inscribed in a wide variety of closed plane figures. This seems a lot easier and more elementary than isoperimetric problems. I realize what I wrote was sketchy, but where do you think it fails? Is there an example where the space of rectangles I described fails to be compact?

[Bullitt]

@LSLGuy

I love these puzzle threads where the actual puzzle finally emerges somewhere around page 2.

But boy do we have fun chasing the other version(s)’ inadvertent rabbit trails. And the rabbit trails of the rabbit trails*. With a few contributions of pure rabbit droppings along the way for spice.

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• Would proper notation for these be rabbit[sup]2[/sup] trails, rabbit trails[sup]2[/sup], or {rabbit trails}[sup]2[/sup]?

Yeah yeah. Admittedly, I deserve that.

Signed,
((Rabbit Trail)^n)^n

[Chronos]

Better than (Rabbit Trail)^(n^n).

[LSLGuy]

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Oh noes! They’re [del]multiplying[/del] exponentiating!

[LSLGuy]

Late edit: I’m happy **Bullitt **took my comment in the spirit I intended. It wasn’t meant as a real dig.

There’s a reason this happens as often as it does. Creating and re-creating correct and unambiguous puzzle definitions is damn hard. It often takes several sets of eyes to spot each of the ways the original exposition is ambiguous, open, or contradictory. English is as fuzzy as math is precise.

And even after all those issues are found and fixed there’s still the question of whether it’s the actual source puzzle or something else.

I’m thinking of that umpteen-page thread a few months ago on the odds of a contingent dice roll game making the rounds on TwitBook where it turns out some popular publisher had made an unwitting “harmless” edit that made the official “right” answer wrong, or at least under-determined, and made one of the “obviously wrong” answers right, or at least the least wrong.

We killed a lot of electrons straightening that one out. Good thing they’re a plentiful natural resource.