16 replies to this topic

[Redezra]

 

Legitimately my favourite things about math. Infinities and fun things to do with them.

[TheSchu]

Infinity is good for things like this.

Too bad the chocolate bar isn't infinite, but it it was, I would not have to stop eating ever!

[Lord Draculea]

Speaking of math games... At least two contestants would be required. Anyone volunteering?

 

You have fifteen seconds. Using standard math notation, English words, or both, name a single whole number—not an infinity—on a blank index card. Be precise enough for any reasonable modern mathematician to determine exactly what number you’ve named, by consulting only your card and, if necessary, the published literature.

 

You can’t say "the number of sand grains in the Sahara", because sand drifts in and out of the Sahara regularly. Nor can you say "my opponent’s number plus one", or "the biggest number anyone’s ever thought of plus one"—again, these are ill-defined, given what our reasonable mathematician has available. Within the rules, the contestant who names the bigger number wins.

 

Are you ready? Get set. Go!

 

 

 

 

 

 

When you've finished, you can read the "right answer" here. 

[He who posts]

Dyscalculia represent.

[TheSchu]

Speaking of math games... At least two contestants would be required. Anyone volunteering?

 

You have fifteen seconds. Using standard math notation, English words, or both, name a single whole number—not an infinity—on a blank index card. Be precise enough for any reasonable modern mathematician to determine exactly what number you’ve named, by consulting only your card and, if necessary, the published literature.

 

You can’t say "the number of sand grains in the Sahara", because sand drifts in and out of the Sahara regularly. Nor can you say "my opponent’s number plus one", or "the biggest number anyone’s ever thought of plus one"—again, these are ill-defined, given what our reasonable mathematician has available. Within the rules, the contestant who names the bigger number wins.

 

Are you ready? Get set. Go!

 

 

 

 

 

 

When you've finished, you can read the "right answer" here. 

The problem with this one is that it is not a math game, but a psych game.

You have to think of what your opponent will think of as a big number and then over top that.

So "the number of sand grains on the earth" is a winner if the other person picks "the number of loaves of bread on earth at the time you read this card".

But if you think of saying "take a plane of the universe at the time you read this card, the number of feet from one edge to the end of the longest axis", the person that uses mm instead of feet wins, and so on.

[Lord Draculea]

The problem with this one is that it is not a math game, but a psych game.

You have to think of what your opponent will think of as a big number and then over top that.

So "the number of sand grains on the earth" is a winner if the other person picks "the number of loaves of bread on earth at the time you read this card".

But if you think of saying "take a plane of the universe at the time you read this card, the number of feet from one edge to the end of the longest axis", the person that uses mm instead of feet wins, and so on.

 

That would be the case if you knew how your opponent might be thinking, but if you have no clue of that, then you're on your own, so you should take your best shot, regardless of what others might come up with. However, by reading the essay indicated by the link, your chances of winning the contest shall increase... how can I put it... exponentially? or maybe... ackermanly? or busy-beaverly? 

[TheSchu]

By the article, the contest is more about notation than imagination.

 

I agree that exponents could give you a large number.

But your judge(s) have to understand the number you provide, so if you use a notation they would not know...

Busy beavers would require you to write out the explanation of the notation in 15 seconds, so I would rule it out along with a great many other similar schemes.

But I would say that skirting the edge of countable infinity (which is what I was doing) would probably ruled out as well, as being too close to infinity

[Lord Draculea]

But your judge(s) have to understand the number you provide, so if you use a notation they would not know...

Busy beavers would require you to write out the explanation of the notation in 15 seconds, so I would rule it out along with a great many other similar schemes.

 

That's where the "reasonable mathematician" and the "published literature" roles would come in. They are mentioned in the contest's conditions. 

 

To quote from the article:

 

[...] even if we can’t list the Busy Beaver numbers, they’re perfectly well-defined mathematically. If you ever challenge a friend to the biggest number contest, I suggest you write something like this:

BB(11111)—Busy Beaver shift #—1, 6, 21, etc

 

The contest is not about notation, in my view. If you know the Busy Beaver concept, any reasonable notation of a BB number will do.

If it was about notation, you could write: BB(BB(11111)), which is a much bigger number than the quoted one. But that would put no new concept on the table. You could even write BB(BB(BB(BB(BB(11111))))), or even a much longer formula, but it would still be based on the same concept. At some point, it would indeed be a challenge to write down a too extended formula (using the same concept) in just 15 seconds. See what I mean? It would be pointless. So, to beat someone who knows about BB numbers would require you to come up with a better concept than the BB numbers. If you can. 

 

The Busy Beavers always win!!! 

[Lord Draculea]

Here's another funny thing about big numbers.

 

Everybody knows what Π is, right? It's an endless non-periodical decimal number, approximately equal to: 3.14159... Let's take the first billion figures of Π: 314159........

 

Question: what's the probability that, by randomly picking an integer number, it contains the sequence of the first billion figures of Π in it?

 

I'm so curious to hear your answers! 

[Redezra]

Zero. It's a finite number of elements in an infinite series (the set of all integers).

[Lord Draculea]

Zero. It's a finite number of elements in an infinite series (the set of all integers).

 

You're half right and half mistaken.    Are you sure you actually meant zero? The question was: what's the probability that, by randomly picking an integer number (from the infinite set of integers), it contains the (finite) sequence of the first billion figures of Π in it?

 

Any other opinions?

[Lord Draculea]

Alright then, I'm going to just throw in the answer.

The probability is 1, or 100%.

How about that?

[Redezra]

Oh cause infinite possible combinations, yeah fair enough.

 

 

Hmmm... that implies there are a countably infinite number of those things.... Difficult to show though.

[Redezra]

Next up, Graham's Number. A number so big that you can't think of a number or concept big enough to explain even the bigness of its bigness.

 

Edited by Redezra, 09 December 2015 - 10:39 PM.

[KiWi]

Numberphile gets my approval.

[Lord Draculea]

Oh cause infinite possible combinations, yeah fair enough.


Hmmm... that implies there are a countably infinite number of those things.... Difficult to show though.

Yeah, our intuitions about infinity are biased, and so is our ability to choose a random integer. We are but tiny creatures in a tiny universe, and our practical experience with numbers is drastically reduced to a tiny neighbourhood of zero. Zero is the Caesar of our world. In practice, it is clear that (almost) nobody will ever pick a number containing the sequence of the first billion figures of Π. However, the mentioned probability (of doing it) is mathematically 1 (or 100%). It doesn't depend of our practical experience or possibilities. I could e.g. say: "an integer in which the sequence of the first billion figures of Π appears for a billion times (not necessarily consecutively)", or "for a billion billions times". The probability of getting such a number when randomly picking an integer is still 1, because the sequence is still finite.

Another funny fact is that the two sets of integers that 1) contains a given finite (though arbitrarily large) sequence of figures and 2) does not contain it are both infinite countable sets. This is very easy to show. For instance, take the series: a₁ = 1; a_n+1 = 10*a_n + 1. The first terms of this series are: 1, 11, 111, ... and so on, and the series is clearly an infinite countable set of integers in which no term contains the mentioned sequence of Π. The fact that the two sets are both infinite doesn't prevent the mentioned probability from being rigorously 1.

 

 

Next up, Graham's Number. A number so big that you can't think of a number or concept big enough to explain even the bigness of its bigness.

Yeah. It's funny though that Graham's number is based on Ackermann's concept. Which is a less powerful concept than the Busy Beaver numbers, in terms of the bigness of numbers they can generate. The Busy Beavers still win!

[Redezra]

And then you have things like the solution to TREE(3), which is so phenomenally large I can't even begin to speak of it ._.