2017-06-29 02:54:36

Two ancient immortal beings float together in the vast emptiness of space... unable to die, and looking for ways to pass the time.  The first titan is the titan of cards.  He has in his possession 3 decks of traditional playing cards (52 cards in each).  The second titan is the titan of dice, and he has in his possession a regular 6-sided die and a less common 4-sided die.  The 2 decide to battle one another, each using their preferred items.

To keep things interesting, each titan comes up with complicated rules to decide who will win each time they play against each other.

It is decided that the first titan will mix all 3 decks together and shuffle them.  3 cards will be drawn.  Each card is given a value depending on what it is, where cards 2 through 10 are worth that many points, jack is worth 11, queens are 12, kings are 13, and aces can be either 1 or 11, depending on what the titan chooses at the time.  For the titan to win, 2 of the 3 drawn cards need to add up to be equal to the value of the last card.  The order the cards were picked in does not matter, it only matters that 2 of the cards add up to be equal to the remaining card.

For example, the titan wins if he were to draw a Queen of hearts, 3 of clubs, and a 9 of diamonds.  The 3 and 9 add up to 12, which is the value of the queen.

The second titan must roll both dice.  If either dice is a 1, the titan loses the round instantly.  If that doesn't happen and both dice are equal to each other, the titan wins.

For example, the titan wins if he rolls 2 and 2, but does not win if he rolls 1 and 1.

Once the titans agree on these rules, they plan to both take turns playing, adding up how many times each of them "wins".  The first titan will draw 3 cards and see if he won, then the second titan will roll his dice and see if he won.  They will repeat this over and over until someone has won a million times!  Take note that every 52 turns titan 1 will have to stop and reshuffle his cards.

The first titan to win 1 million times will be declared the ultimate winner, and will be in charge of controlling the radio for the next hundred years as they float through space.  Which titan will win?

- Aprone
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2017-06-29 03:49:42

For me the dice one.

2017-06-29 05:30:31

Great. Now I have to math. Thanks aprone. LOL!!! Okay, the titan 2 has the easiest solution. A simple sample space will tell us that titan's odds of winning once. Now, to be clear, titan 1 has 156 cards, but as it draws, the number of cards will go down until 52 cards have been drawn, then the titan will be back to 156? I also take it that the titan doesn't necessarily need to win the game 1000000 times in a row?

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C++: The object-oriented programming language of a pagan deity" -- The Red Book
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2017-06-29 06:42:53

Jase, titan 1 draws 3 cards each time, so after 52 plays he will have no cards left.
You are correct... they don't have to win all those wins in a row.  Just a total.

- Aprone
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2017-06-29 06:47:37 (edited by CAE_Jones 2017-06-29 06:50:26)

.(I recommend voting before reading replies smile )

Ehehehaa, tricky. The real trick is how the Titan of Cards's chances change, while the Titan of Dice's stay the same. It would seem that, in every round of 52, the Titan of Cards has better odds for the first 12 draws, but worse for the remaining 40. I keep failing to memorize the (n^2-n)/2 thing and keep having to rederive it (maybe this time it will stick?). Now you're telling me to integrate a probability curve tongue.
I'll take the easy way out and assume that having worse odds for 10/13 of the time means the Titan of Cards loses on the long game.(I recommend voting before reading replies smile )

.(I recommend voting before reading replies smile )

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2017-06-29 07:16:04

Wait, if titan 1 has 3 decks, and makes one deck of 52 cards, then, is this a standard deck of 52, or is there a possibility of duplicates?

I have a website now.
"C: God's Programming Language
C++: The object-oriented programming language of a pagan deity" -- The Red Book
"There, but for the grace of God go I"

2017-06-29 08:21:29 (edited by stewie 2017-06-29 08:25:34)

The card titan wins, you can't roll dice in space.

Although, you could I suppose rig the dice as the dice titan. If they are trying to be fair, then I think the card titan wins. If not, the dice titan would win each time. Then again if they were truly fair they'd pick games that both sides are possible to play without cheating.

Deep in the human unconscious is a pervasive need for a logical universe that makes sense. But the real universe is always one step beyond logic.

2017-06-29 09:31:44

Jase, it is 3 standard 52-card decks combined into a single 156-card one.  The new combined deck would contain 3 of every card.  Since the suit of each card doesn't really matter, that could be ignored and the deck would contain 12 of every number.  For example, there are 12 queens in his deck (3 queens of hearts, 3 queens of clubs, ect).

Stewie, your dice rolling in space is definitely worth a thunk, but this puzzle assumes the titan found a way to roll them.  big_smile

CAE Jones, I suppose I am sir.  LOL.

- Aprone
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2017-06-29 09:55:36

Something about the phrasing: "The second titan must roll both dice.  If either dice is a 1, the titan loses the round instantly."

How is a round defined? Is it defined to be the set of one turn from both titans? or the whole 1000000 or more sets?

Deep in the human unconscious is a pervasive need for a logical universe that makes sense. But the real universe is always one step beyond logic.

2017-06-29 10:27:14

Reading back I can see how I should have perhaps worded things a bit differently to avoid confusion.

Titan 1 draws 3 cards and sees if he won, meaning 2 of his cards' values add up to equal the remaining card.  If he happens to have won, he keeps track of his victory since he is trying to rack up a total of 1 million wins before titan 2 can do the same.

Now Titan 2 rolls his dice and sees if he won, meaning he rolled doubles but not 1-1.  If he happens to have won, he also keeps track of his total victories.

Now that each titan has gone, this cycle repeats and it is again Titan 1's turn to draw 3 more cards.  Hopefully this helps clear things up a bit.  As soon as either titan reaches 1 million accumulated "wins", everything stops and that titan is declared the ultimate winner.

- Aprone
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2017-06-29 10:32:47

Oh ok, I just wanted to make sure that titan 2 rolling two 1's meant it didn't lose the entire game.

Deep in the human unconscious is a pervasive need for a logical universe that makes sense. But the real universe is always one step beyond logic.

2017-06-29 16:42:56

So far the votes are leaning pretty far in one direction.  I'm not sure if people have been able to work out the odds for each titan, or if people are relying on their gut feelings and intuition.  In either case I'm just happy to see 14 people have bothered to read my crazy little post so far, haha!

- Aprone
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2017-06-30 12:55:36

Titan of cards clearly wins, while I think his odds are rather worse over all, he does have the power of paper cuts as a last resort where as the four sided   die has only soft plastic corners and the six sided one will do no good at all.
so, when they get tired of the statistics and resort to violence, Cards guy definitely  wins big_smile.

With our dreaming and singing, Ceaseless and sorrowless we! The glory about us clinging Of the glorious futures we see,
Our souls with high music ringing; O men! It must ever be
That we dwell in our dreaming and singing, A little apart from ye. (Arthur O'Shaughnessy 1873.)

2017-06-30 17:23:57

Well it has been a few days so I should wrap this riddle up.  80% (16 people) sided with the dice titan, and they were correct.  Well done!

The dice titan's odds of winning are 12.5%
The card titan's odds of winning are about 12.21%, so very very close, but he still has no chance of being the first to 1 million wins.

- Aprone
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2017-06-30 21:08:38

Yeeei!

2017-07-01 06:16:00

So now that you gave the answer, can you explain how you get the card titan's odds? I didn't vote because I couldn't figure it out.

2017-07-01 10:25:25

The dice odds were pretty straight forward, but those cards odds are definitely tricky.  To be honest, I didn't figure out how to work it out mathematically... I just simulated it instead.

I threw together a little program that underwent the whole process of drawing 3 cards from the 3 combined decks, and checking the rules to see if it was a winning draw or not.  It repeated the steps over and over, seeing what percentage of the time worked out to be a win.  Repeat a few hundred million times and you get a very accurate approximation of the winning odds.  (so I cheated)  big_smile

- Aprone
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