Card Game for my Discrete Structures class

[crAZcarl]

So my teacher is making us write a paper on various games he proposed to us. This is the one I choose.

You get 5 random cards from a standard deck. You place 4 of them in any order you like face up, and the last one face down. Knowing your strategy of placing the first four cards, someone could figure out what the face down card(suit and value) is.

So here's my strategy:
Since there are 5 cards and only four suits, there will be atleast 2 of one suit. So you put those two cards first and last. Since there are only 13 cards in the deck, you can reach any value from any other value by adding 1,2,3,4,5,or 6 to the starting value. (A queen(12) plus 4 would be 3..it loops around). So for the other three cards you have, there are 6 ways to arrange them, which is exactly what we need. You will arrange them according to their value. If two cards have the same value, refer to their suit.(Alphabetically... Clubs < Diamonds < Hearts < Spades)
Low, Mid, High = 1
Low, High, Mid = 2
Mid, Low, High = 3
Mid, High, Low = 4
High, Low, Mid = 5
High, Mid, Low = 6
So with this strategy we can do any 5 cards in the standard 52 card deck.

Example:
2s 5d 7c 9h Qh

Put the 9h First, then you need to add three, so Mid, Low, High
9h 5d 2s 9h _??_

And then knowing our strategy someone else could tell us that 9h plus the three will be the Qh

Now comes the real part. I need to start adding Jokers into the mix.
So for one joker, I see four scenarios:
1. no joker
2. joker and two other cards w/ the same suit.
3. joker and one of each suit(but not all the same value)
4. joker and one of each suit all w/ same value.

Now here's my strategy
1. Use regular strategy
2. "Lie" to our partner. Tell them that the card facing down is a card that is already on the board(in the three cards in the middle) and they will know that it is a joker.
3. Use the joker as the highest card possible in the three middle cards to determine value, and know that the last card is the suit that is not already shown in the 4 cards
4.Place the joker first, and know that the last card is the suit not shown already, and the value of the three cards in the middle.

So I can get all that, but now my teacher tells us that it IS possible to add two jokers.
Cases
1. No jokers
2. 1 Joker and 2 of the same suit
3. 1 Joker and not 2 of the same suit
4. 1 joker and 4 of the same values, all different suits
5. 2 jokers and 2 of the same suit
6. 2 jokers and not 2 of the same suit
7. 2 jokers and 3 of the same values, all different suits

Strategy(This is the part that I can't figure out)
1. "Lie" to the user and have the last card be joker1(you must differentiate between joker1 and joker2. If you get joker 2, use it in the three cards as the highest card, with first and last suit matching.
2. You can use joker1 in the middle three cards, with the suit not showing being the suit that is hidden. I don't know what to do with joker 2. I know I can't use it in the middle again.
3-7 all depend on #2, so I'm pretty much stuck here.

Our teacher also recommended looking at the combination and permutations of the 52 card deck with respect to the 4 and 5 cards, but I'm not too familiar with what those mean exactly.

Thanks for any help/ideas/suggestions you can give me. If anything needs clarification let me know(I'm usually pretty scatterbrained when writing these things, but I tried to be as clear as I possibly could).

 

[mighty2000]

Dear fellow scientist,

Thank you for sharing your chosen game and your strategy for it. It is a very interesting and challenging game indeed, and I can see why your teacher chose it for your paper.

After reading through your strategy, I believe I have some ideas and suggestions that may help you with the part you are stuck on - adding two jokers into the mix.

Firstly, as your teacher suggested, let's look at the combinations and permutations of the 52 card deck with respect to the 4 and 5 cards. This means looking at all the possible ways of choosing and arranging 4 or 5 cards from a deck of 52.

For 4 cards, there are 52 choose 4 possible combinations, which is equal to 270,725. This means there are 270,725 different ways to choose 4 cards from a deck of 52.

For 5 cards, there are 52 choose 5 possible combinations, which is equal to 2,598,960. This means there are 2,598,960 different ways to choose 5 cards from a deck of 52.

Now, let's look at the permutations. This means looking at all the possible ways of arranging the chosen cards. For 4 cards, there are 4! (4 factorial) possible permutations, which is equal to 24. This means for each of the 270,725 combinations, there are 24 different ways to arrange the 4 cards.

For 5 cards, there are 5! (5 factorial) possible permutations, which is equal to 120. This means for each of the 2,598,960 combinations, there are 120 different ways to arrange the 5 cards.

So, how does this help us with adding two jokers into the mix? Let's look at each of the 7 cases your teacher mentioned and see how many combinations and permutations there are for each.

1. No jokers - This is the simplest case, and we already know there are 270,725 combinations and 24 permutations for 4 cards, and 2,598,960 combinations and 120 permutations for 5 cards.

2. 1 Joker and 2 of the same suit - In this case, we have 51 cards to choose from (since one card is the same suit as the joker). This gives us 51 choose 3 possible combinations, which is equal to 208,250.