Interesting things

My Solution to the Collatz Conjecture

As promised, here’s my attempted solution to the Collatz Conjecture. My solution is pretty simple, so if you understand the conjecture, you should understand the proof. (I’m not a pro mathematician anyway, just an amateur hobbyist.) I’m eager to get feedback, especially if I somehow missed something subtle (or worse, something really stupid).

PDF of my proof: click here.

If you prefer to watch a video instead, I’ve uploaded myself explaining my solution here:

Here’s to hoping my proof is confirmed!

By S P Hannifin, ago

A problem of chains

As you’re traveling the road, it’s a strange and frightening thing to look around and suddenly realize that most of the world is not walking with you, that what you believe is right and wrong is actually not at all what most of the world accepts or lives by.

At first, it makes the world seem so sad and dark, almost post-apocolyptic, despite the smiles on everyone’s faces. How can I be a part of this world? I can’t live by these standards.

But then it doesn’t seem so bad. You’re just looking at the world the wrong way. You are the enlightened. It’s everyone else that will eventually discover the sad darkness of their world; they’re only smiling because they’re looking down most of the time. They haven’t reached the ends of their chains yet, so they don’t yet realize that they are slaves. But you are not a slave. You know how to look around for chains, and you can’t be chained if you’re always on the look out for them.

Oh, how vain this all sounds! Do I think I’m better than everyone else?

Not intrinsically, and certainly not everyone. They all have the capacity to shed their chains if they want to anyway. But the chains feel good. I’ve been in chains before. I admit that I unfortunately still even let them slip on from time to time. But without them, I can certainly see more than many others. I can understand more. If that sounds vain or prideful, so be it. I am not fooling myself into thinking I can see everything, after all. But I am not going to say that what I see might not be there just because so many others can’t see it. I am not going to disbelieve my eyes for the sake of people who aren’t even looking where I’m looking.

But isn’t it arrogant to think of them as slaves?

No. Why should it be? It would probably seem arrogant to try to help them out of their chains by saying: “Hey, look around! You are a slave!” That wouldn’t help.

And it would be prideful to presume that I’ve got all my chains off, wouldn’t it? Maybe there are chains I can’t see, or chains I’ve let slip back on without noticing. But I can’t deny the existence of chains just because of this. Nor can I stop caring about them. At least I am aware of them and can work to get my own chains off.

Isn’t this a bit wacko? Haven’t I met others who have warned me about chains that are foolish to believe in? Don’t some people see chains where there are none? Yes. They see some real chains, and then they mistake so many other things for chains. “Your shoes are chains!” “Your glasses are chains!” No, they’re not. I think they are mad. And don’t those who don’t see chains at all think I am just as mad? Is there any way to deal with that?

Perhaps not. I can only be honest with myself.

So what about the chains of others? Should I do anything about them? Can I do anything about them? What relationship am I to have with this world that I think is so dark and sad? I certainly can’t force people out of their chains. If only it were so easy. Chains can only be removed by the person in them. Should I ignore the chains of others and just encourage them to look around for them by my own chain-free actions? Perhaps that is the best way. Perhaps the only way. And if they never get out of their chains, so be it. I cannot blame myself for it.

But I still struggle with what I should feel about them. In their chains, they do things that hurt me and each other. They kill (“that’s not a person” or “he deserves death”), they imprison unjustly (“we must force each other to do things”), they lie and steal (“he doesn’t need this; I should have it”). I often feel inclined to hurt them back. But that is the tug of a chain I shouldn’t be bearing. But they don’t have to be in chains! I am so angry with them!

Should I feel sad that they are in chains, or happy that I am not?

I don’t think sadness will get anyone anywhere, as natural as the reaction is. As long as I keep vigilant with my happiness, that is the path I must stick to. If I am happy that I am not in chains, then at least I am keeping the chains away. It is hard, because the chains do slip on sometimes and are comforting, and tearing them off can be painful at first. But the happiness possible without them is a clearer happiness.

A constant struggle, but the road always ends in light, and it will be easier to travel if I keep that in mind, if I am looking forward and not backward.

By S P Hannifin, ago

The forgetting problem

Here’s a question I made up (though I’m sure someone’s asked something like it before… nihil novum sub sole). I don’t know the answer to it, I’m still trying to puzzle it out:

Say you’re trapped in a science lab. The scientists there have invented a way to erase 24 hours worth of memory each night at midnight.

You have several choices on how to spend your day. You may either:

1. Commit suicide
2. Be tortured half the day then play for the other half
3. Be tortured the entire day, giving you the privilege to play all day the next day

Regardless of your choice, you will continue to be stuck in the lab and continue to have your memory erased each night until you die.

The question is not about what you would choose. The question is: does it matter?

UPDATE: Though one could come up with a bunch of variants, what about a guilt-based one? What if you were stuck in the same lab, had the same forgetting-at-midnight scenario going on, only this time the choices are:

1. Be tortured while another science lab prisoner does not suffer for that day
2. Do not suffer while another science lab prisoner is tortured all day

For #2, you might feel guilt, but is that better than the torture itself, especially considering you’ll both completely forget the experience the next day?

Gah, this problem is really annoying me…

By S P Hannifin, ago

The crime-twins problem

I had a weird dream the other night. (It was random and dream-like, like dreams often are, but for the purposes of this post I’ll pretend like it was more understandable than it really was; the gist of it is the same.) I was the judge of a no-jury court case. It involved a man who had stolen priceless paintings. All of his illegal activities of breaking and entering and stealing these paintings from a museum were all recorded by security cameras. His face could be seen very easily.

Unfortunately he had an identical twin brother. They were both sitting there in the court room before me, both claiming innocence, both with the same solid alibi of walking around a bookstore that night, which was also caught on camera. They both knew exactly what one of them was doing at the bookstore. I questioned them separately (I’m not sure if that’s really something a judge would do, but I did), and they both told the same story, which supported what was on camera.

There was no DNA evidence left behind by either man.

Lastly, I did not have the lie detecting skills of that guy in Lie to Me, so I couldn’t do anything very dramatic involving catching one of them in a lie or giving them the Prisoner’s Dilemma.

And in the dream I thought it was a riddle, as if there definitely was a way to solve the conundrum and I just had to figure it out.

But I woke up with the riddle unsolved.

Is there an answer? Or is that the perfect way to get out of a crime?

The dream made me wonder if DNA would really help such a case anyway. It might! According to this article from Scientific American:

The discovery of this genetic variation gives hope for an obscure but pressing issue in the case of a criminal suspect who is an identical twin. “If one twin is a suspect and the whereabouts of the other twin cannot be determined, then the jury is often left without the ability to find guilt beyond a reasonable doubt” in cases that rely on DNA evidence, says Frederick Bieber, a pathologist at Harvard Medical School.

“If the twin issue comes up in a criminal investigation it’s possible that if there are [copy number variants] that differ between the two twins that might help sort that out,” Bieber says.

Given that there are 80 pairs of identical twins in Virginia’s convicted offender database alone, this might not be as small an issue as it may sound. And such genetic variation also matters to the population at large.

Ha! Not a very original idea for a dream at all, I guess. Subconscious fail.

By S P Hannifin, ago

Little probability problem

OK, here’s a pretty simple probability problem, but it took me a moment to think about… until my geniusness came up with the solution. The problem is:

You want to choose randomly between 3 choices, but all you have is one coin. What’s the minimum amount of times you need to flip the coin so that you can choose fairly between your 3 choices? (That is, so that each choice has a 33.3…% of being chosen?)

The first person to answer correctly gets the prize of being the first person to answer correctly.

(To complicate the problem, suppose you have two coins, but one has been tainted with iocane powder, and no immunities allowed…)

By S P Hannifin, ago