Coin Game puzzle, part 1 Name: Take some coins and arrange them in piles. You may make any number of piles and have any number of coins in each pile. The piles may be different heights and a pile may have as few as one coin in it. Now, from that starting arrangement pick up the top coin from each piles and make one new pile out of those coins. Notice that any pile that had just a single coin was eliminated. You now have a new arrangement of the coins. For convenience we list the piles in non-increasing order by size. Repeat this process to the new arrangement of coins, i. e. pick up the top coin from each new pile and make one new pile out of those collected coins. Keep repeating the process. Although this process never ends, weâll call it quits if we get back to a state (arrangement of coins) weâve already been. A state that is eventually revisited is called cyclic and the number of steps from a cyclic state to its next repeat is called the length of the cycle. A 1-cycle state is called a fixed point, i. e. a fixed point is unchanged by the process. (a) Perform the process beginning with the given state and record the progress until you revisit a state.
(2,1,1,1)
(1) (3,3)
(4.2.1) (2,2,2,2)
(b)
i. Give an example of fixed points from your earlier work.
ii. Give several different examples of fixed points.
iii. Make & conjecture about which states are fixed points under this process.
iv. Make a conjecture about the number of coins in a fixed point state.
(c)
i. Start with an arrangement of ten coins, perform the process, and list the progression.
ii. Start with an arrangement of ten coins that you haven't encountered yet, perform the process, and list the progression. You may stop when you reach a state listed in part (i).
iii. Start with another arrangement of ten coins that you haven't encountered yet, perform the process, and list the progression. Stop when you reach a state listed in part (i) or part (ii).
iv. Make a conjecture about how the process with ten coins will always end.