12. given four coins with tails up, determine how many moves it will take to get all coins to show heads if…

Explanation:

Step1: Initial state

Let's represent tails as 0 and heads as 1. Initially, we have four 0s (all - tails), and we want four 1s (all - heads). Each move changes the state of three coins.

Step2: First move

Suppose we turn three coins. The number of heads changes. Let's consider the parity (even - ness or odd - ness) of the number of heads. The initial number of heads is 0 (an even number). When we turn three coins (an odd number of coins), the parity of the number of heads changes.

Step3: Analyze the parity requirement

To get all four coins to show heads (4 is an even number), we note that each move changes the state of three coins (an odd - numbered change). Let (x) be the number of moves. The total number of coin - flips is (3x). We want (3x\equiv4\pmod{2}). Since (3\equiv1\pmod{2}), we have (x\equiv0\pmod{2}) (because (3x) must be even to get from an even number of initial heads (0) to an even number of final heads (4)).

Step4: Find the minimum number of moves

We can try to find the minimum number of moves by brute - force. First move: Turn three coins. Now we have 3 heads and 1 tail. Second move: Turn the three non - flipped coin (in the first move) and one of the previously flipped coins. Now we have 2 heads and 2 tails. Third move: Turn three coins again. Now we have 1 head and 3 tails. Fourth move: Turn the three non - flipped coin (in the third move) and we get 4 heads.

Answer:

4