Safe Search 2.8.1. A Cirele of Coins. Alice and Bob play a gar alternate moves,

Question

Safe Search 2.8.1. A Cirele of Coins. Alice and Bob play a gar alternate moves, and Alice starts. There is a cirele of 12 coins, and on each move a player can remove either one coin or two coins that were originally adjacent on the circle. For example, on Alice's first move she might remove the coin at position 3, as shown in Figure 5. Then Bob might remove the consecutive pair of coins at positions 8 and 9: but Bob may not remove the two coins at positions 2 and 4, because they were not originally adjacent. Figure 5. Alice and Bob's first two moves.

Explanation / Answer

Answer:

There is 12 coins in the game. Here the question is player who removes the final coin wins.

We can think that Bob will win if he prepares the symmetry. So if Alice picks 1, then Bob will pick only one coin. However to separate and symmetric arcs of coins are left and if he picks 2 then he will remove 2 coins to gain.

Let us consider for n coins in this case.

Assume Alice is the one who is making the first move

If n=1 or 2, Alice is going to win

If n=3, Alice is never going to win

If n=4, Alice wins if he picks one at first.

Here we can look into other angle.

Bob wins if he picks two at first

If n=5, Alice wins if she picks 2 at first.

Here we can observe that for Alice to win because she is making the first move to not let the symmetry being created.

But here if symmetry is getting created usually Bob will win everytime.

Thank you!

Related Questions

Navigate

• Browse (All)
• Browse S
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.