What if you had one queen frog and one lazy frog? Is it always possible? If not, when?
Answer: Yes, its always possible. Since we know that we can work with the queen frog with any destination, we can split the line around the lazy frog, and solve for two stacks which can come on top of the lazy frog
What if there are some lily pads that are waterlogged - i.e. no frogs can get there?
Answer: It seems that none of these configurations can be solved for, but there isn't a proof yet
Horses have four legs. Behind, they have two legs, and in the front, they have forelegs - ad infinitum!
In the game of Palindrome Grab, a string of red and black checkers are on the table in a straight line. On your turn, you take checkers from either the left or right end, provided that the collection of checkers you take is a palindrome - in other words, symmetric, same backwards and forwards. Of course one checker is always a palindrome.
For example, if the checkers are RBRRB, from the left you could take R or RBR, and from the right you could take B or BRRB, so you have four choices for your move.
If there are 3 or more checkers, what's the least number of checkers you can always take?
Answer: 2 checkers are always possible
If there is a long string of checkers, what's the least number of checkers you can always take - is it still 2, or is it much larger? Does it depend on the length of the string?
Answer: The answer is still 2. Consider RRBRBRBRBR....BRBRBB
What if you have an odd number of checkers?
Answer: Still 2. Consider RRBBBRBRBRB...RBRR, except for 5 and 7 strings which require 3
Greedy Game - The winner is the one who gets the most checkers
Who wins with RBRBBBRRBBBRBRB, player 1 or 2?
Who wins with RRBRRBBRBRRRBRBBRRBRB?
Play a few more games between groups of two to get familiar with the game?
Is this game deterministic, i.e. given any configuration, and assuming smart play from each person, it can be predicted who will win (or draw)?
Can you explain how to construct a long string where player 1 wins?
Answer: BRBRBRBR...BRBRBR
Can you explain how to construct a long string where player 2 wins?
Answer: BBRBRBRBRB...RBRBRR
Patient Game - The winner is the one who gets the last checker, regardless of how many each person has collected.
Play a few games between groups of two to get familiar with the game?
Can you explain how to construct a long string where player 1 wins?
Answer: BBRBRBRBRB...RBR - Take first B and second player will leave an odd length string
Can you explain how to construct a long string where player 2 wins?
Answer: BRBRBRBR...BRBR - Player 2 always gets odd length string
Homework
Are there strings where player 1 wins the greedy game but player 2 wins the patient game?
Answer: BRBRBRBR...BRBR
Are there strings where player 2 wins the greedy game but player 1 wins the patient game?
? Does RBRRBRBRBR...BRBB work? (player 1 taking the first R as opening move)