Well, there have been many interesting enigmas for thousands of years.

The oldest one and most known is Sphinx's Enigma.
(In fact, in ancient Greece, even oracles and mathematical requests were
made in a form of an enigma, to stimulate creative thinking!)

Here are two recent logical ones posted here in the past two days:
viewtopic.php?p=231642#p231642
viewtopic.php?f=7&t=19093&start=0

As for paradoxes, there are aplenty too, and there is even one mechanical,
also known as Ferguson's Paradox. I also have a "puzzle device" of it:



So, give here your best shot of an enigma, riddle or paradox!

Oh yes, and did I tell you that I AM A LIAR?




Pantazis

Well, since you didn't specify that you are a pathological liar who always lies 100% of the time, there's no paradox.

How many grains of sand does it take to make a heap of sand?

Also, the gear paradox thingy looks amazing! Where could I get one to fool my science teacher with?

Here's a VERY simple one:
A khalifa dies and leaves his 11 camels to his 3 sons, for the oldest one half the camels, for the second a quarter of the camels, and for the youngest one sixth of the camels, since the idea wasn't exactly cutting camels they had a big problem.
A very wise old man came by on his camel and told them, "if your father would've left 12 camels it would be easy for you to divide them, here, take mine, do the division and I won't lose anything"
On what does the old man base his assumption?

And 10 more:
http://listverse.com/2007/09/21/top-10-brain-teasers/

Even simpler one: Opposite Day

If it's Opposite Day then that means it's not Opposite Day. But if it's not Opposite Day then it's Opposite Day. But if it's Opposite Day then it's not Opposite Day. But if it's not Opposite Day then it's Opposite Day. But if it's Opposite Day then it's not Opposite Day. But if it's not Opposite Day then it's Opposite Day. But if it's Opposite Day then it's not Opposite Day. But if it's not Opposite Day then it's Opposite Day. But if it's Opposite Day then it's not Opposite Day. (More nonsensical babbling)

Now waiting to see an Oskar / Houlis / Ferguson Gear Cube.


Well if this reply is true then you didn't even ask that question.

I love the Ferguson device! It's a great one for party tricks. "Hey, the gears aren't turning together" Not overly expensive (for those of us that weren't there) well worth the money if you can get one.

Panda, have you got a video of the Ferguson thingie in action?

the below post is telling the truth

the above post is lieing

I love riddles, so here's one: A lady is reading a book at her house. When she gets tired she turns off all the lights and goes to sleep. The next day she reads in the newspaper that a boat crashed killing 400 people, and she then comminted suicide. Why?

Turns out, Pantazis, I know of one problem that all of us but you can reason about correctly.

Consider: "Pantazis cannot consistently believe this sentence."

If he believes it, then he is being inconsistent, in believing that he can't believe it. Therefore it is true. By this logic, all the rest of us can see that it is true, and believe it, but not poor Pantazis.

Here are some famous related paradoxes.

Berry's Paradox: What is the smallest positive integer not nameable in fewer than twelve words? (Hint: "the smallest positive integer not nameable in fewer than twelve words" contains eleven words.)

Russell's Paradox: Does the set of all sets that don't contain themselves contain itself?

spoiler (highlight):

she lives in a lighthouse

Thank you Gödel . I love self-referential logic -- but not as much as self-referential sentences .

Since Russell's paradox has been thrown out there, I feel like I should add the Banach-Tarski paradox into the mix, for fans of mathematics here on TP. Of course, neither of these are really paradoxes, Russell's paradox is only a paradox to philosophers who don't really understand how axiomatic set theory works, and the Banach-Tarski paradox is just a true statement which intuitively seems false if you've never been exposed to measure theory. Anyway, the Banach-Tarski paradox:

Consider the unit sphere in 3 dimensional space. It is possible to divide the sphere up into finitely many pieces (you can get away with only 5 pieces, in fact), and reassemble the pieces ("reassemble" here essentially means what it should mean -- only rigid motions, i.e. rotating and sliding) into TWO copies of the unit sphere. Of course, if that's not weird enough for you, you can iterate this process as many times as you want and, say, turn a sphere the size of a golf ball into a sphere the size of Jupiter by cutting it into finitely many pieces and rearranging them finitely many times.

Write this down and ask someone to answer both true-or-false questions. Most people start confidently, then begin having trouble when they re-check their answers!



They can't both be true because if the second sentence was true then the other could not be. But if the second sentence is false then the first sentence must also be false, because otherwise the second sentence would be true.

The paradox is seemingly resolved if the first sentence is false and the second is true...but most people dislike that option!

Well, Russell's Paradox was certainly a mathematical paradox at the time it was discovered; that's one reason there is axiomatic set theory! And the Banach-Tarski Paradox (which is truly wonderful) is only a true statement if you use the axiom of choice. Some mathematicians use this fact as an argument against the axiom of choice!

Right, of course. Thank goodness for modern set theory. I guess pretty much both of these are best understood as tidbits of mathematical history than anything else. Nowadays, who doesn't use AC? ZF without choice has some pretty strange models.

While we're at it, if people know what models are, how about the Skolem paradox -- by the Löwenheim–Skolem theorem, ZFC has a countable model, which is a countable set that satisfies all the theorems of ZFC, such as, say, "there are uncountably many real numbers"... Also not a paradox, per se, but it's fun to grapple with when you're still getting used to working with models.

My favorite paradox is Zeno's "Achilles and the Tortoise" paradox.

Text from Wikipedia:
In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 meters. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run 100 meters, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say, 10 meters. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise.

I've never been happy with any of the resolutions that either prevent time from being infinitely divided or don't allow an instantaneous snapshot of time.

LOL I was hoping someone would mention set theory. For those who are into comics,
the "Logicomix" is something I would recommend them to read. It features Russell
and his paradox, while it goes through all of the trouble until... (ok, I won't spoil it for you!)
Some sort of preview here: http://www.youtube.com/watch?v=XebglmXrgEc
To find it, just search amazon or ebay, there are many translations in different languages!

Oh, and by the way, people who think that they are absolutely sure about something are silly.
(I am 100% sure about this!)




Pantazis

This one is easy, and has always upset me. He's not racing to where the tortoise was, he's racing to the finish line. Just like you can't ever touch something because you have to go half the distance to it, then half that distance, then half that distance. Solution? Go PAST the object you want to touch. All of a sudden half the distance between where you're going and where you are is what you really wanted in the first place.

Well, if I am lying 100% all of the time, how come this time, in the sentence
above, I am actually telling the truth about me?






I think I have a video of it somewhere, but I have not publicised it yet.
It is not really anything spectacular, and many gifs can show a better
demo of the paradox (which was not really a paradox, but it was just
a trick to confuse some watch-makers LOL).




Pantazis

I think that's sort of punting on the real problem, by saying "if you look at it this way instead, it works fine". True enough, but what's wrong with the original line of reasoning that seems paradoxical?

The problem is that the paradox assumes that it will take forever to reach the infinite number of points Achilles must reach to catch the tortoise. But that's simply not true; that series (100 + 10 + 1 + .1 + ...) converges when you sum it (at 111 1/9). You don't have to prevent time from being infinitely divided or disallow an instantaneous snapshot of time. You just need more powerful mathematics than the Greeks had available. They assumed, implicitly, that no infinite sum of finite numbers could be finite. Which turns out to be false.

Here's one I enjoy.

A genie offers to sell you a magic lamp at a price you choose. If you buy the lamp, you get 3 wishes for anything (except to break the second rule). After you get your 3 wishes, you must sell the lamp to someone else cheaper than what you bought it for. So if you bought it for $5.00, you would have to sell it for $4.99 or cheaper. If you can't sell it, your soul will be damn and you be tortured for all eternity. The person you sell the lamp to has the same rules apply to them.

Now here's the paradox:
Obviously, nobody would ever buy the lamp for 1 cent because there no smaller amount of money you could sell it for, and your soul would be damned.
Again, obviously, nobody would buy the lamp for 2 cents because nobody would buy it from them for 1 cent, and your soul would be damned.
Again, obviously, nobody would buy the lamp for 3 cents because nobody would buy it from them for 2 cent, and your soul would be damned.
etc, etc... By induction, you should never buy the lamp.

But people will almost always chose to buy the lamp because they assume that they can find someone else to buy it.

My favorite enigma: MEN

@GuiltyBystander: that's an easy one, because there's no time limit to sell it, so you can buy it for 1 cent and keep it until after you die, by which time you don't care about being tortured.

That's also an easy one: we actually say what we mean.

Very nice. Here is a similar example of irrational human behavior, that can be quite entertaining.

What you do is, auction off a dollar to the highest bidder. But there's a catch: not only does the highest bidder pay his bid, but the second highest bidder also pays his bid, even though he gets nothing. So, what happens? People will generally start bidding around a nickel. Hey, why not, if you might get a dollar for a nickel? But as the price inches higher, a new dynamic takes over: the second-place bidder doesn't want to lose his bid. So if he's bid 60 cents and it's at 65 cents, he will likely bid 70 cents (hoping to net 30 cents vs. losing 60). It becomes a game of chicken. If you think it's irrational to bid higher, then you can reason that your bidding opponent will not outbid you, which justifies your bidding higher!

But here's where it gets crazy. This same dynamic keeps going, well past a dollar! I know of cases where a dollar has been auctioned this way for over $20. Which means $40 profit for the auctioneer. People have to be pretty stubborn for it to reach this level, but if you have enough people together, you can almost always sell a dollar for more than a dollar.

This is a well-known effect of auctions called "winner's curse".

"Pyrrhic victory" also sounds appropriate.




Pantazis

Lol. I'd love to see that.

Would you believe that this amazing sentence contains ninety two letters, one comma and a single question mark?

This computer-generated pangram contains six A's, one B, three C's, three D's, thirty-seven E's, six F's, three G's, nine H's, twelve I's, one J, one K, two L's, three M's, twenty-two N's, thirteen O's, three P's, one Q, fourteen R's, twenty-nine S's, twenty-four T's, five U's, six V's, seven W's, four X's, five Y's, and one Z.

LOL nice!

Allow me to expand to another form of sentences (palindromes) which involves
symmetric letters and photos of a sympathetic animal which usually poses beside
the puzzles made by many builders of this forum:

WAS IT A CAT I SAW?




Pantazis


PS. Of course, palindromes may be made in other ways, which involve words
instead of letters etc etc etc

Alex

This page has six images with palindrome humor.
(The more I stare at the one with the cow the more it makes me laugh!)




Pantazis

Is the answer to this question "no"?

I'm lying right now.

- Greg

World's longest palindrome (so far): http://norvig.com/pal17txt.html

But... it doesn't make any sense.

"Dammit I'm Mad"

Dammit I’m mad.
Evil is a deed as I live.
God, am I reviled? I rise, my bed on a sun, I melt.
To be not one man emanating is sad. I piss.
Alas, it is so late. Who stops to help?
Man, it is hot. I’m in it. I tell.
I am not a devil. I level “Mad Dog”.
Ah, say burning is, as a deified gulp,
In my halo of a mired rum tin.
I erase many men. Oh, to be man, a sin.
Is evil in a clam? In a trap?
No. It is open. On it I was stuck.
Rats peed on hope. Elsewhere dips a web.
Be still if I fill its ebb.
Ew, a spider… eh?
We sleep. Oh no!
Deep, stark cuts saw it in one position.
Part animal, can I live? Sin is a name.
Both, one… my names are in it.
Murder? I’m a fool.
A hymn I plug, deified as a sign in ruby ash,
A Goddam level I lived at.
On mail let it in. I’m it.
Oh, sit in ample hot spots. Oh wet!
A loss it is alas (sip). I’d assign it a name.
Name not one bottle minus an ode by me:
“Sir, I deliver. I’m a dog”
Evil is a deed as I live.
Dammit I’m mad.

Demetri Martin wrote this 224 word palindrome poem.

Funny you should bring that up... I'm one of the eight people that independently came up with that sentence, as a result of a challenge in Scientific American in the mid-80s. It took quite the computer search, at the time.

The challenge was posed by Lee Sallows, who had built a custom computer to do a brute-force search to construct a similar sentence, stepping through all possibilities. It took months of run time to find a solution. He was so proud of his accomplishment, he declared that it couldn't have been accomplished with just software, and challenged anyone to prove different. Several of us did. Exhaustive search for this kind of problem is just stupid.

Yes, but the effect is highly magnified when you make the 2nd-place bidder pay too.

Today's xkcd is highly relevant:

i'm not going to post in this thread

I say, you are right.
You say, I am wrong.

Indeed the Banach-Tarski Paradox is a very interesting case.

But we need to consider something important: density.

Yes, density. It may not make sense for infinite sets, but it is the
"rate of distribution of points" which changes.

Something like thinking that [4 in the power of "infinity 1"], "goes faster"
to infinity than [2 in the power of "infinity 1"]. Note that I used the term
"infinity 1" to ensure that the powers used are of the same "rate".
(i.e. [4 in the power of "infinity 1"] could be equal, lower or higher than
[2 in the power of "infinity 2"], depending on the relation between
"infinity 1" and "infinity 2")

I mean, since we have defined the term "density" for the finite cases,
we are required to do the same for the infinite cases. And if we don't,
then, as seen, many paradoxes may pop up.

That said, the properties of the infinity density *do not* have to be
the same with the finite one. The starting point would be defining the
simplest of all infinities. This simplest infinity has neither the lowest nor
the fastest "rate", but it can be a nice starting point.

Finally, for the "density" to make sense, we also need a "point of reference".
For example, when we compare two infinities from the same set (e.g. fractals)
it *does* matter how deep we have gone. The 1-1 and onto homomorphism
(i.e. isomorphism) seems naked without the notions of "density" and
"point of reference". Yes, it works perfectly for finite isomorphisms, but
isomorphisms between infinite sets are by definition more complex and
demanding, so a more careful approach and a new theory is required.




Pantazis


PS. I love this talk, I almost drove crazy one of the Professors when I was
undergraduate, and when I showed to him some "new" theory, he could not
tell me the reason it was right or wrong. LOL

We can safely say two things.

1. A million grains of sand is a heap of sand.
2. We can take one grain of sand out of the pile, and it will still be a heap of sand.

So, if we continue to take grains of sand out of the pile, we end up with one grain of sand. Is this a heap of sand? Better yet, if we take away the last grain of sand, is the nothingness remaining still a heap of sand?

Not if a heap of sand is 1,000,000 grains as you defined previously. Then it's 999,999 grains, which is one short of a heap.

Yeah, this 'paradox' has always irritated me. You can't specify what a heap is and then later choose that you decide by looking.

If 1,000,000 is an arbitrary number though, you are deciding that it's a heap just by observation, so every time you remove a grain you should decide whether it's a heap or not. Of course it would be boring, and people would have different interpretations of what a heap is, but nobody would ever reach 1 or 0 as suggested.

Alex

Everything somehow leads to 4 in the end.
for example:

92. Ninety-two has 9 letters
nine has 4 letters.

But not only numbers equal 4 in the end, even things
Like:
Drewseph's Chopasaurus. Those two words are made up of 20 letters
Twenty is made up of 6 letters
Six is made up of 3 letters.
Three is made up of 5 letters.
Five is made up of 4 letters
and there's 4 again.

Can you come up with a word, thing, or number that does not lead to 4?

3.1415926535897932384626433...

Or what about $, €, £, and ¥? (the symbols, not what they represent )

In my opinion, everything ends with g.