I hope you appreciate my attempts to show interesting aspects of mathematics that can also be useful. But here among the “number games” are some really funny stories.
1) The 14 Earths
The story began as follows: On a whim (we, that is my partner of many years and still a good friend, Michael), we wanted to construct a backgammon position in which it was highly improbable that one side would win. But it should still be possible.
You should know that every position in which there is still contact, i.e. in which it is still possible to win, even if only theoretically, is always much more probable than you can imagine. This is often in the percentage or per mille range.
The position we constructed looked like this: there was no contact at all. One then also speaks of a “pure race”. Whoever rolls higher wins. Of course, one side can have a lead, in which case it is enough to roll the dice just a little lower and you still win. Depending on the size of the lead, the chance can of course be very large to gigantic. Curiously, I often see, even among very good players, that in the final phase of the game, when the last pieces are rolled, they start to calculate whether it is possible that they can still win. They then count the stones and calculate. Does that make you feel smarter somehow when you have it out? I think it’s a waste of time. I just roll the dice until all the stones are out, mine or the opponent’s. And if he thinks I’m a loser because of that, I’m a loser. And if he thinks I’m a weaker player because of that, then that wouldn’t even be a disadvantage.
So in the position we constructed, if the trailing party rolled doubles 6 10 times and doubles 4 (or even doubles 6) once in the following 11 rolls, they could still win. Tiny side condition: The opponent, the one in front, had to roll 1-2 at the same time 11 times. If he rolled higher once, it was guaranteed to be a win. These two probabilities multiplied together give a chance of 1.57*(10 high -24). So a probability of about 1.57 to 1000 000 000 000 000 000 000, written out.
Then Micha was immediately ready to risk 1 DM. He wanted to play the leading party and I should play the outsider. He would pay me the correct odds if I won.
I briefly considered the possibilities, it was only a joke anyway, I could have given him one DM or bought him a coffee. So, just for fun, I said, “OK, we can do that. But you have to put down the equivalent amount first.” It’s called putting it on the table, so to speak.
Now it was up to him again to think. Then we simplified the matter and thought about it together. It really got out of hand. And once we started, there was no stopping us. We calculated. We estimated an infinite number of variables, of course. In the process, we flowed through the powers of 10. In the end it came out that if he had wanted to “lay” this amount, he would have had to lay 14 earths. But the peculiarity: All of them would have had to be filled to the brim with 1000 DM notes.
Maybe that would have exceeded the table’s carrying capacity?
2) Nine little numbers… explode the universe
As an exercise, I once thought about this: Put the nine digits, i.e. all the digits from 1 to 9, together so that they add up to the largest possible number. Then I thought about doing it as a competition. And then you start checking the entries. There is a prize for the best solution, the largest number.
What concerned me much more than the question of how to put the numbers together in the best possible way was the problem of checking the entries. Because you would have to calculate each number first, and that could go beyond any computer capacity. Does that sound strange? So let me explain:
The optimal solution from my point of view looks like this: 2 ^ 3 ^ 4 ^ 5 ^ 6 ^ 7 ^ 8 ^ 91. The biggest problem is to place the 1 in such a way that it causes the least damage, because any number to the power of 1 or times 1 or whatever does not change the number. And if you use it in a smaller power, it never becomes that effective.
So if you compare 2 ^ 3 ^ 41 with 2 ^ 31 ^ 4 then 2 ^ 3 ^ 41 is much bigger. In fact, to judge the size of the number, the sequence of arithmetic operations must be taken into account. So the number 2^(3^4) is larger than (2^3)^4, where (2^3)^4 = 4096 and 2^(3^4) is larger by a ridiculous 18 powers of ten, namely = 2.41785 * 10^24. My Excel can even handle the number 2^(3^6). This results in the ridiculous trifle of 2.824 * 10^219. So with these little three digits I have produced a number that has a length of 219 digits. It is no longer pronounceable anyway. At least it can still be represented.
But if you now imagine the 6 remaining digits and the explosion that occurs with the modification of the one small digit (from 4 to 6 as a power, I achieve an increase of almost 200 powers of ten), then you can perhaps imagine the explosion that occurs with a further digit that is also a power. Just for the sake of completeness: The correct notation for the (so far) largest number with the 9 digits would actually be 2^(3^(4^(5^(^6 (7^( 8^91))))))).
I then tried, just in my mind, to switch dimensions and imagine how to make this number approximately describable. And to do this, I imagine the string of zeros that follows the leading 1. Only it is advisable not to represent this in a chain at all, but rather in volume. So I write zeros box by box, house by house, earth by earth. Always one zero per cubic millimetre. Don’t say they’re too big. But even if they are, smaller doesn’t help.
Since I have now laboriously processed the first three digits, i.e. the 2^(3^4), but can’t cope with the 5 at all, other means of comparison have to be used. The final 8^91 is already a 1 with 82 zeros. But that is only the exponent for the number before it, which is already exploding all dimensions. And even if one should succeed in accommodating the zeros of the number up to 7 in our Milky Way, then this number raised to the power of a 1 with 82 zeros is guaranteed to be the end of our universe, it is bursting at the seams.
I can only imagine the evaluator who then has to tell the lower-placed candidates that their solution is too small by a trifle 628 million universes compared to the winning solution. And all I did was string a few digits together, just 9…
The first chess book I opened began with an anecdote about how the inventor of the game proposed the game to his Shah (yes, Persia, hence the word “chess”), who was so enthusiastic about it that he wanted to grant the inventor a wish. He said he would only wish for a grain of rice on the first square, double the number on the second, then double the number again, i.e. 4 on the third and so on. The Shah was so touched by the modesty of the wish and thought to fulfil it without further ado. Unfortunately, he soon had to realise that the entire stock of rice on earth would probably not be enough. Clever people calculated much later the length of the chain of trucks that would have been required to transport all these grains of rice. This would supposedly have spanned the earth several times.
Well, that may sound impressive, but I can still write down the number 2^64 without further ado. That is equal to 1,844 * 10^19. You could even pronounce it. That’s just 18.44 trillion. Sounds a bit like our budget deficit.
On the other hand, I always try to draw attention to the luck factor in chess. Almost all players (happily) ignore this. All the good moves that someone makes for the wrong reason, but which turn out to be effective and good, are attributed either to intuition or as “planned”. That almost everyone fibs in the process is part of chess. Vanity (have I already talked about this a little?) is a faithful companion (be careful what you say, Paule!) of people with diminished self-esteem. However, this is very widespread, there are hardly any exceptions.
Other factors of luck such as the “condition of the opponent”, “accidental, unplanned development of a position”, “gross, not corresponding to the opponent’s level, loss-making blunder of the opponent”, “lucky constellation of the pieces” or also “despite overlooking the countermove, it just goes well”, apart from “luck of the draw in the tournament” or also “mistakes of the competitors in decisive games without one’s own participation” and several others are benevolently ignored. You have a position, you know all the legal moves, you have the possibility of decisively influencing events by your own choice of moves.
That’s why I always like to present to any doubters my original reasoning for proving luck: If you teach a monkey (that could well be a computer) how to play chess and it knows nothing more than (know) all the legal moves in a position and then has to choose one at random from all the legal moves, it would also have the possibility of executing all the moves that Kasparov would also have made at that moment. The probability may seem small. I have assumed that a game lasts 50 moves on average and that 22 legal moves are possible per position, which results in the number 22^50. This gives a probability of 1 in 1.3* 10^67. This number can no longer be pronounced, but it can still be easily represented. So with a probability of this value, the monkey has now executed all Kasparov’s moves.
But then, in the position where Kasparov’s opponent gives up, that would certainly not be advisable for the opponent. After these 50 moves, which the monkey would then have executed like Kasparov, the monkey would still lose 99.9999999999999% of all games afterwards. Because even if he had a whole bishop or even rook advantage (which in grandmaster practice would be guaranteed to lead to resignation, even minor advantages), if he continued to choose moves randomly he would still lose the game, practically always.