## CD. Alas, Up To A Million Dollars Might Have Been Given Away.

We discuss the results of the fabulous Math Factor Million Dollar Giveaway. and confess this was an excuse to bring up Game Theory and how to talk about really big numbers.

1. ### greggk said,

March 20, 2007 at 8:24 pm

Speaking of large numbers…I was thinking about combinatorics recently and came up with an interesting number. The podcast had mentioned numbers like a googolplex were not too useful because they exceed the number of particles in the universe – but I often wonder about such large numbers in term of combinations such as this one:

256^(640×480)

Can you guess what this refers to? You might be looking at an example right now.

Think about all the pixels on a computer monitor and the possible color values of each one. Now – how many images can that monitor display? An infinite number, right? A monitor with 8-bit color with a resolution of 640 x 480 would have this many total discrete screen displays or images. (If I did my math right) Could we program a computer to generate the entire set of images? I supposed you could prove that there is not enough matter in the universe to create enough hard drives to store all these images. Although 99.99% of these possible screen images would appear to be random noise, a subset of those images would include an image of … everything. It would include a photo of every person who had ever lived — and also the same photo but wth the person wearing say… a Yankee cap. And it would include every frame of film or video ever shot …and its negative. Its hard to imagine the total number of screen imagines is really not infinite – but I think its no bigger than 256^(640 x 480), right?

And you can use use any type of monitor you want of course. My little digital casio watch is 120 x 120 pixels back and white if I recall correctly- so I guess that’s 2^(120×120) – still a nice sized number.

2. ### strauss said,

March 29, 2007 at 2:27 pm

It’s kind of amusing to consider how many possible states the universe could be in! That comes in much bigger than a googolplex, but not so big at all compared to some of the numbers we’ll be discussing soon!

Lessee, very very roughly (everything is only to within a few dozen orders of magnitude) The number of possible positions in the universe is, well, now wait a minute. I guess let’s be generous and say that two positions are different if they are at least 10-33 m apart. That’s a lot smaller than any particle, but is apparently considered the smallest meaningful scale. And let’s just suppose the universe is a box 1010 light years on a side, which comes to about 1078m3 in volume. So there are, say 10177 different positions possible.

This is totally ridiculous, so please don’t jump all over any inaccuracies!

Now there are something like 10 80 particles in the universe, according to something I saw someplace. Somewhere else I think I might have heard there are 10125; since we are just aiming for a rough sense of things:

Let’s then suppose there are 10200 positions and 10100 particles. This gives a whopping (10^100)^(10^200) possible states for the universe:

10^10^202. Let’s just call it an even 10^10^200, shall we.

This is one big number.

3. ### isenham said,

March 27, 2007 at 1:43 pm

I am a high school teacher in southern California. Yesterday I gave my AP Calculus students a quiz titled, “Up to 1000 points (or more) Quiz.” Similarly to the Million Dollar Giveaway, the winner was the student who submitted the largest real number and the extra-credit points earned were equal to 1000 divided by the winning entry. The students had fun with it. The largest entry submitted was 10^38 written with a lot more zeros. The smallest entry submitted was 10^-22. Obviously this students was hoping to never worry about a quiz again.

4. ### isenham said,

March 28, 2007 at 12:44 pm

I forgot to mention that one of my students gave a very clever reply on the quiz. His answer was, ” the number A such that A is one more than the largerst number chosen by any other student.” Should I delcare him the winner?

5. ### strauss said,

March 29, 2007 at 2:09 pm

I don’t knowâ€” seems a little cheap to me! At least he specified that his number should be larger than any other student’s! (It’d be sad to see him disappear down a logical rabbit hole!)

6. ### Sye Heinlein said,

March 30, 2007 at 12:29 am

In reply to the first post.
WOW. There is software that can recognize faces and some other objects.Imagine if they could build a quantum computer that is capable of firstly generating these images and secondly analyzing them to see if they are faces.
That would be pretty cool and hard to get your head around.
Of course I don’t think even quantum computers would be able to carry out all these operations within a single lifetime.

7. ### contactm3 said,

June 7, 2007 at 3:16 pm

I know it’s a bit late, but I came with a huge number which I never submitted. It comes from a sequence I put together, A(x), where:
A(1)=1, A(2)=2^2, A(3)=3^3^3, A(4)=4^4^4^4, A(5)=5^5^5^5^5, A(6)=6^6^6^6^6^6, A(7)=7^7^7^7^7^7^7, etc. My first number, which I call a “Nikiplex” (inspired by the name of my girlfriend) = A(100). My second number, the one I was going to submit, which I called a “Nakiplex” = A(1 Nikiplex). This is substantially larger than “the number of possible states of the universe” ^ “the number of possible states of the universe.” Perhaps one day there’ll be another large number contest and I will become the winner of the smallest monitary prize ever. Until then, great job to these participants!

8. ### strauss said,

June 7, 2007 at 3:31 pm

Ah, very nice indeed; in fact, though, as truly immense as they are, these numbers are STILL kind of small compared to some of the entries we had!!

In the discussion of Graham’s Number we talk about Knuth’s “arrow notation”. In this notation, A(n) is written n^^n, and one Nikiplex is 100^^100. A Nakiplex is (100^^100)^^(100^100)

This is way smaller than 100^^100^^100^^100 (the order of operations is read right to left) which is just 100^^^4 in the arrow notation. That arrow notation is hard to beat!!

But then Graham’s number trounces that hugely, by iterating arrows, much like you were iterating exponentiation, over and over again.

The REALLY STAGGERING THING though, was Rayo’s number, which is so frighteningly immense as to defy description. In fact, that is its definition. It is the smallest number bigger than any number that can be described in fewer than a googol’s worth of symbols, in any mathematical manner. To put that in perspective, a Nakiplex took fewer about 20 symbols in the Knuth notation; Graham’s number can be described in fewer than 100. If we list out EVERYTHING that can be mathematically described in fewer than a googol’s symbols, and then top that, we get Rayo’s number.

(Of course I just described Rayo’s number in about a paragraph, but the notion of “mathematically describing” has a very precise meaning.)

YOW!

9. ### contactm3 said,

June 12, 2007 at 2:47 pm

WOW! I withdraw my earlier statement! I grossly misunderstood Knuthâ€™s â€œarrow notationâ€. I mistook 10^^10 as meaning 10^10^10, when it really means 10^10^10^10^10^10^10^10^10^10. Wow! That Knuth guy had big things on his mind. Then to iterate arrows… whoa. Good job Rayo! (I don’t know how I’m going to break it to my girlfriend).

10. ### strauss said,

June 26, 2007 at 7:05 am

This email arrived this morning, to the mathfactor address. I can only assume it is related to our own fabulous million dollar give-away!

Euro Millions Lottery SPE.,
2011 Zaventem , Belgium .
Euro Millions are Affiliate of Belgium National Lottery (BNL).

CONGRATULATIONS: YOU WON â‚¬1,000,000.00

We are pleased to inform you of the result of Euro Millions Lottery SPE, which was held on the 9th June. 2007. You were entered unaware as an independent email participant with: Ticket Number: 657-954-2509 with Serial Number-1413-09. Your email address attached to Lucky Draw number: 4-11-17-23-30 with Bonus number 25 which consequently won the Euro Millions Lottery SPE lottery in the 3rd category. This lucky draw came first in the 1st Category of the Sweepstake. You will receive the sum of â‚¬1,000,000.00 (One Million Euros) only from our authorized bank.

Because of some mix-up with sweepstake prizes, including the time limited placed on the payment of your prize:,

etc etc etc

Just as lucrative as our give-away, I’m sure.

11. ### dfollett76 said,

July 31, 2007 at 8:49 pm

I think the problem with this contest was we all really knew the point before we submitted numbers, and we didn’t really believe there was \$1 million dollars available. I’d like to see this experiment on a non-math-enthused audience where \$1 million dollars is really available. Perhaps on the audience at a game or talk show.

12. ### strauss said,

July 31, 2007 at 9:42 pm

Interestingly, a wide majority of the entries were less than 20,000; the median was roughly 1000.

Sure, we expected (and hoped for) a few cynics to wreck things— but plenty of people tried for a real prize!

I think the outcome would be about the same when played with any large group. Smaller groups, especially small groups of acquaintances, have a very different dynamic.

13. ### contactm3 said,

August 2, 2007 at 6:02 pm

I’m curious how a group’s “submission median” would respond to a small tweak in the rules:

“The winner gets \$X + (\$1,000,000/submission)”

I imagine the submission median would rise without bound (after first having slowly approached, then departed from 1,000,000/X). I think Rayo would have walked away with the \$20. :)

Also, for smaller groups of, lets say N acquaintances, I wonder how the group’s “submission median” would respond to the following tweak in the rules:

“The winner gets \$1,000,000/submission, but each loser must provide 1/Nth of the prize money to the winner”

Just food for thought.