Ok, so I just finished a Discrete Math class where we learned about all types of proofs and counting and graphs, etc. So I know the basics, but never thought I'd need to use it in the real world (not anytime soon at least). However, a friend of mine showed me a puzzle game, and he said that it's always possible to win this game, even though it seems unlikely to me. I will explain the rules, which are pretty simple, and I want to know how I could prove this, or if the answer is evident using combinatorics.
Here's how the game works: The game is a slight variation of a game called 'Poker Squares'. Anyway, you lay down (all at once) 25 random cards arranged in a 5x5 grid (no jokers). The goal of the game is to arrange all the cards so that each row is a poker hand of the following:
- A straight: 5 cards arranged in order of their number (i.e. 9,10,J,Q,K, irregardless of their suite.)
- A flush: any 5 cards in any order of the same suite (i.e. five hearts)
- A full house: A triple and a double (i.e. three 10's and two J's.)
- A straight flush: 5 cards of the same suite arranged in increasing or decreasing order (i.e. 4,5,6,7,8 all diamonds)
And thats it. The claim is that given any random 25 cards, these cards can ALWAYS be arranged so that each row is a winning poker hand (of the above mentioned hands only).