How to avoid being hit by a laser in a room of mirrors [video]

https://www.youtube.com/watch?v=jJ6FD59U0_E

[Red Circle]: I suggest a Duel of Titans, "mano a mano" the only true test for gentlemen.

[Valet leading Green Circle to room of mirrors]: If you kill him, all this be mine. Monsieur, Good shooting.

Although it does also rather make me think of Buster Keaton standing still as half a house falls over him and the cameraman closes his eyes.

https://ericlippert.com/2011/02/14/what-would-feynman-do/

I spent a whole month making friends with the guards, only to have them change shift at the last minute. Thankfully, my cohort pointed out I could wear a reflective suit. All is well that ends well!

https://www.youtube.com/watch?v=jJ6FD59U0_E

But perhaps instead of 4^2 blockers, you'd need 4^3. But, I'm making guesses based on intuition, which makes for bad math.

- in a square room all angles are right

- with wall at right angles the laser light will always create an inscribed square and return to the source for any given angle

- there is exactly one angle in a range <0,90> at which the light bounced from a single wall will go through an arbitrary point in the room

- there are 4 walls which gives at most 4 squares to be blocked

- the blocking point can be set anywhere on the inscribed square before the target - this gives at most 2 points per square (we can consider them left- and right- -hand directed) hence 8 points necessary

- at start we selected only angles from the <0,90> degree range for the ease of calculation, so there are also symmetric versions in the (90,180> range

- which gives us at most 16 points to block all the possible squares in both directions

These are two separate problems.

This example doesn't show all places where the shadow is.

It only shows, when you place yourself and a light, an example arrangement of occluders that will prevent you from being hit.

https://youtube.com/shorts/DyRPASsynhI

For one thing, in my experience, working on real life problems involves a whole lot more "memorizing problems" than working on mathematics.

And thank goodness for that! When I drive over a bridge or install an app, I really don't want to hear that whoever made it has an aversion to memorizing problems and solutions. Solving a real world problem is usually boring. Making a real life thing well is usually about correctly putting together many pieces that some other people have put together, in a way that's roughly similar to how somebody else has already put a lot of those pieces together before. Most of the work is well-trodden and uncreative.

I think part of the reason mathematics feels like it's about "memorizing problems" is that getting something deeper than that out of it is a habit/skill. A very important aspect of the vague notion of "mathematical maturity" is a habit of looking at a solution to a problem with the mindset of "how could I have come up with this?". That is, unpacking a problem and solution into some deeper understanding or way of thinking that led to it. As opposed to filing the problem and solution away "as is" into some toolbox to be referenced later. A lot of "gotcha!" solutions to mathematical problems are unsatisfying precisely for this reason.

In a lot of cases once you've read a problem and read a solution to that problem, and understood the solution, you've done about 10% of the work. The remaining 90% is this difficult work of unpacking and repacking lessons from that solution into something that actually deepens your understanding of what the problem is about. Sometimes reading the solution is actually doing negative work.

In other words, if you look at this problem and look at this solution and think "neat, but I don't get anything out of it beyond 'neat'", that's normal and fine, and probably correct. But that doesn't mean there isn't anything in it beyond 'neat', and a big part of learning mathematics well is to dig deeper than that even when the problem and solution don't force you to. Whether that digging is worth it is up to you. (It's often kind of a crapshoot in terms of payoff.)

But that's a more complicated situation than "this shows a fundamental flaw in mathematics".

And similarly, whilst one proof might have limited practical applications, it's very difficult to know which proofs do have practical applications and it's almost impossible to know which results might later yeild further proofs that do have practical applications.

None of this is really a matter of intelligence, it's a matter of putting in work (although obviously there is intelligence involved to find particularly neat ways of appraoching problems).

And why do mathematicians learn proofs? Often, to see the approaches that you can use to solve other problems.