Chocolate Bar
Half-eaten chocolate
You have a bar of chocolate as above. The bottom left piece is dark chocolate and no one likes dark chocolate, right?
Your mother wants you to share the chocolate with your brother, so you come up with an idea. Each turn one person can pick a piece and gets to eat everything to its right and above. The person who ate the dark chocolate loses. You are to play the first move, what is the sum of all piece that can guarantees you a win?
Assume that your brother plays optimally.
Clarification
- If you choose the piece, you get to eat the piece .
The answer is 8.
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1 solution
The only piece that guarantees you a win is piece 8. (Take piece 8. If your opponent takes piece 1, take piece 3. If your opponent takes piece 2, take piece 7. If your opponent takes piece 3, take piece 1. If your opponent takes piece 7, take piece 2.)
All other pieces will result in a loss, assuming your opponent plays optimally. (If you take pieces 4, 5, 6, 9, 10, 11, 12, or 13, then your opponent can take piece 8, and use the same strategy outlined above against you.) If you take piece 1, then your opponent takes piece 12. (What is your opponent's strategy then?) If you take piece 2, your opponent takes piece 7. If you take piece 3, your opponent takes piece 9. And if you take piece 7, then your opponent takes piece 2. In all these cases, you will ultimately lose.