12. given four coins with tails up, determine how many moves it will take to get all coins to show heads if…
12. given four coins with tails up, determine how many moves it will take to get all coins to show heads if turning three coins at a time counts as one move.
Answer
Explanation:
Step1: Represent coin - state
Let 0 represent tails and 1 represent heads. Initially, we have four 0s (0, 0, 0, 0) and we want to get to four 1s (1, 1, 1, 1). Each move flips three coins.
Step2: Analyze the parity
The total number of coins is (n = 4). When we flip three coins at a time, consider the sum (S) of the states of the coins (mod 2). Flipping three coins changes the parity of (S). Initially, (S=0) (since all coins are tails). The target state has (S = 0) (since (1 + 1+1 + 1=4\equiv0\pmod{2})).
Step3: Make moves
First move: Flip three of the four coins. Let's say we get (1, 1, 1, 0). Second move: Flip two of the heads - up coins and the tails - up coin. We get (0, 0, 0, 1). Third move: Flip the three tails - up coins. We get (1, 1, 1, 1).
Answer:
4