Is there any way to account for the fact that certain seemingly simple problems tend to yield answers that seem rather complicated?

Take this problem for instance:

https://i.imgur.com/nGPLOAj.png

Wolfram Alpha comes up with this answer:

https://imgur.com/a/bul0AZc

An interactive demo for this problem:
https://www.desmos.com/calculator/oplzocbf26

On 9/7/2023 1:02 PM, sobriquet wrote:

> Is there any way to account for the fact that
> certain seemingly simple problems tend to yield
> answers that seem rather complicated?

I don't have an answer beyond merely
pointing out that that is how it is.

I see it as a result of
things not always being how they seem.
If that's an explanation,
it's not much of one.

Meta-complicating things,
the complicated-ness of answers as
a function of time trends downward.
We might, occasionally,
find less a complicated answer to
the same question.

(A counter-trend is our tendency to
address ever-more-difficult questions.
But I'm holding the question constant.)

> Take this problem for instance:
> https://i.imgur.com/nGPLOAj.png
> Wolfram Alpha comes up with this answer:
> https://imgur.com/a/bul0AZc
> An interactive demo for this problem:
> https://www.desmos.com/calculator/oplzocbf26

Nice.
Here, I think the reason that the answer is
so complicated is that it is the solution
to a quartic polynomial.

https://i.imgur.com/nGPLOAj.png
a square inside a quarter-circle

A (x,0) intersection of one corner, x-axis
B (0,y) intersection of second corner, y-axis
C (y,x+y) intersection of third corner, circle
D (x+y,x) fourth corner
E (x+5,0) intersection of x-axis, circle

C and E are the same distance from (0,0)
D and E are 2 units apart

|C|² = |E|²
|D-E|² = 2²

y² + (x+y)² = (x+5)²
(5-y)² + x² = 4

y² + (x+y)² = (x+5)²
2y²-25 = 2x(5-y)

(5-y)² + x² = 4
4(5-y)⁴ + (2x(5-y))² = 16(5-y)²
4(5-y)⁴ + (2y²-25)² - 16(5-y)² = 0

L² = x²+y²

On Thursday, September 7, 2023 at 10:53:59 PM UTC+2, Jim Burns wrote:
> On 9/7/2023 1:02 PM, sobriquet wrote:
>
> > Is there any way to account for the fact that
> > certain seemingly simple problems tend to yield
> > answers that seem rather complicated?
> I don't have an answer beyond merely
> pointing out that that is how it is.
>
> I see it as a result of
> things not always being how they seem.
> If that's an explanation,
> it's not much of one.
>
> Meta-complicating things,
> the complicated-ness of answers as
> a function of time trends downward.
> We might, occasionally,
> find less a complicated answer to
> the same question.
>
> (A counter-trend is our tendency to
> address ever-more-difficult questions.
> But I'm holding the question constant.)
> > Take this problem for instance:
> > https://i.imgur.com/nGPLOAj.png
> > Wolfram Alpha comes up with this answer:
> > https://imgur.com/a/bul0AZc
> > An interactive demo for this problem:
> > https://www.desmos.com/calculator/oplzocbf26
> Nice.
> Here, I think the reason that the answer is
> so complicated is that it is the solution
> to a quartic polynomial.
>
> https://i.imgur.com/nGPLOAj.png
> a square inside a quarter-circle
>
> A (x,0) intersection of one corner, x-axis
> B (0,y) intersection of second corner, y-axis
> C (y,x+y) intersection of third corner, circle
> D (x+y,x) fourth corner
> E (x+5,0) intersection of x-axis, circle
>
> C and E are the same distance from (0,0)
> D and E are 2 units apart
>
> |C|² = |E|²
> |D-E|² = 2²
>
> y² + (x+y)² = (x+5)²
> (5-y)² + x² = 4
>
> y² + (x+y)² = (x+5)²
> 2y²-25 = 2x(5-y)
>
> (5-y)² + x² = 4
> 4(5-y)⁴ + (2x(5-y))² = 16(5-y)²
> 4(5-y)⁴ + (2y²-25)² - 16(5-y)² = 0
>
> L² = x²+y²

Yeah, that's the approach they took in the youtube video for the problem:
https://www.youtube.com/watch?v=7ZIImuW0PBA

Just specify some algebraic equations and then wolfram can easily provide
a numerical solution, and often also an exact solution (the desmodder plugin
will allow you to easily transfer such huge expressions from wolfram alpha
to desmos).

It's still fun to just play with it interactively though, to get a feeling for how
the biggest possible square can rotate within the confines of a quarter circle (by
using the slider for variable c in the desmos demo).

On Thursday, September 7, 2023 at 10:53:59 PM UTC+2, Jim Burns wrote:
> On 9/7/2023 1:02 PM, sobriquet wrote:
> >
> > Is there any way to account for the fact that
> > certain seemingly simple problems tend to yield
> > answers that seem rather complicated?
> >
> I don't have an answer beyond merely
> pointing out that that is how it is.

*lol*. That would have been my answer too.

| "It's just the way it is." :-P

On Thursday, September 7, 2023 at 11:34:24 PM UTC+2, sobriquet wrote:

> It's still fun to just play with it interactively though, to get a feeling for how
> the biggest possible square can rotate within the confines of a quarter circle (by
> using the slider for variable c in the desmos demo).

Yeah, though the "corresponding math" seems to be rather "cumbersome".

Nice example, thank's for that!

On Thursday, September 7, 2023 at 10:03:04 AM UTC-7, sobriquet wrote:
> Is there any way to account for the fact that certain seemingly simple problems tend to yield answers that seem rather complicated?

Probably there are more 'simple problems' than there are 'simple answers'

On Thursday, September 7, 2023 at 11:34:24 PM UTC+2, sobriquet wrote:

> It's still fun to just play with it interactively though, to get a feeling for how
> the biggest possible square can rotate within the confines of a quarter circle (by
> using the slider for variable c in the desmos demo).

Yeah, though the "corresponding math" seems to be rather "cumbersome".

Nice example, thanks for that!

torsdag 7 september 2023 kl. 19:03:04 UTC+2 skrev sobriquet:
> Is there any way to account for the fact that certain seemingly simple problems tend to yield answers that seem rather complicated?
>
> Take this problem for instance:
>
> https://i.imgur.com/nGPLOAj.png
>
> Wolfram Alpha comes up with this answer:
>
> https://imgur.com/a/bul0AZc
>
> An interactive demo for this problem:
> https://www.desmos.com/calculator/oplzocbf26
I'm not sure why you are so surprised. It's easier to state a problem than to solve it, obviously.

On Friday, September 8, 2023 at 8:21:32 AM UTC+2, markus...@gmail.com wrote:
> torsdag 7 september 2023 kl. 19:03:04 UTC+2 skrev sobriquet:
> > Is there any way to account for the fact that certain seemingly simple problems tend to yield answers that seem rather complicated?
> >
> > Take this problem for instance:
> >
> > https://i.imgur.com/nGPLOAj.png
> >
> > Wolfram Alpha comes up with this answer:
> >
> > https://imgur.com/a/bul0AZc
> >
> > An interactive demo for this problem:
> > https://www.desmos.com/calculator/oplzocbf26
> I'm not sure why you are so surprised. It's easier to state a problem than to solve it, obviously.

I think somehow it conflicts with the popular idea of the "unreasonable effectiveness
of math". It might indicate that perhaps there are alternative ways to conceptualize it
that yield more compact symbolic expressions.
Just like if you try to do stuff like multiplication or division in the Roman
number system it gets kind of awkward for bigger numbers, while switching to
the Hindu-Arabic number system enables you to deal with it more
effectively.
Hypothetically speaking, if you had the Roman number system and you didn't know
about the alternative Hindu-Arabic number system, you might assume that doing
multiplication or division of numbers is just complicated and hence there is no reason to
assume it might be possible to come up with other number systems that simplify things.

On Friday, September 8, 2023 at 5:23:19 AM UTC-7, sobriquet wrote:
> On Friday, September 8, 2023 at 8:21:32 AM UTC+2, markus...@gmail..com wrote:
> > torsdag 7 september 2023 kl. 19:03:04 UTC+2 skrev sobriquet:
> > > Is there any way to account for the fact that certain seemingly simple problems tend to yield answers that seem rather complicated?
> > >
> > > Take this problem for instance:
> > >
> > > https://i.imgur.com/nGPLOAj.png
> > >
> > > Wolfram Alpha comes up with this answer:
> > >
> > > https://imgur.com/a/bul0AZc
> > >
> > > An interactive demo for this problem:
> > > https://www.desmos.com/calculator/oplzocbf26
> > I'm not sure why you are so surprised. It's easier to state a problem than to solve it, obviously.
> I think somehow it conflicts with the popular idea of the "unreasonable effectiveness
> of math". It might indicate that perhaps there are alternative ways to conceptualize it
> that yield more compact symbolic expressions.
> Just like if you try to do stuff like multiplication or division in the Roman
> number system it gets kind of awkward for bigger numbers, while switching to
> the Hindu-Arabic number system enables you to deal with it more
> effectively.
> Hypothetically speaking, if you had the Roman number system and you didn't know
> about the alternative Hindu-Arabic number system, you might assume that doing
> multiplication or division of numbers is just complicated and hence there is no reason to
> assume it might be possible to come up with other number systems that simplify things.

Like in other technologies, the biggest part of any solution is getting the appropriate tools and machinery set up. And the hours/weeks/years/centuries that it took to develop those tools and machinery should be counted when determining the simpleness/complicatedness of a solution.

The proof of the Fundamental theorem of Galois theory is only a few lines. But it takes a couple semesters before you can understand enough to comprehend the proof.

I saw a thread on facebook a while back where some parents were complaining about how their children and grandchildren are being taught to do long division. A common statement was that this method was so much 'harder' than the method WE had been taught. Forgotten was the hours/weeks/years it took us to learn 'our' method. And, incidentally, totally unnoticed was the fact that with the new method you have to do exactly the same guesses, multiplications, subtractions, and additions that WE do using OUR method.