Franziska and Safua play the game Halvsies. The game is set up with one pile of...

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Franziska and Safua play the game Halvsies. The game is set up with one pile of 5 stones, one pile of 8 stones, and one pile of 11 stones. Franziska, who's younger than Safua, always goes first. On their turn, a player does one of the following: They may remove stones from a single pile. They remove at least one stone and no more than half of the pile (rounded up). They may split one pile, consisting of more than one stone, into two new piles as evenly as possible. Whichever player removes the last stone is the winner. Of course, Halvsies could be played starting with any number of piles and any numbers of stones. Answer the following questions to generate some avenues of exploration and perhaps solve the original game. (a) Show that if the game were set up with two piles of equal size, Safua can guarantee a win. Conclude that if they played Halvsies with a single pile with an even number of stones, Franziska can guarantee a win. (b) Suppose Safua wins in a single-pile game with A stones and also in a single-pile game with B stones. Does she win in the game which starts with one pile of A and one pile of B? (c) Suppose Franziska wins in a single-pile game with A stones and also in a single-pile game with B stones, and A B. Does she win in the game which starts with one pile of A and one pile of B? (d) Hopefully there are some questions that naturally arose that you can pursue now. Can you determine who wins in other games of Halvsies? Who can guarantee a win in the original game set-up? Franziska and Safua play the game Halvsies. The game is set up with one pile of 5 stones, one pile of 8 stones, and one pile of 11 stones. Franziska, who's younger than Safua, always goes first. On their turn, a player does one of the following: They may remove stones from a single pile. They remove at least one stone and no more than half of the pile (rounded up). They may split one pile, consisting of more than one stone, into two new piles as evenly as possible. Whichever player removes the last stone is the winner. Of course, Halvsies could be played starting with any number of piles and any numbers of stones. Answer the following questions to generate some avenues of exploration and perhaps solve the original game. (a) Show that if the game were set up with two piles of equal size, Safua can guarantee a win. Conclude that if they played Halvsies with a single pile with an even number of stones, Franziska can guarantee a win. (b) Suppose Safua wins in a single-pile game with A stones and also in a single-pile game with B stones. Does she win in the game which starts with one pile of A and one pile of B? (c) Suppose Franziska wins in a single-pile game with A stones and also in a single-pile game with B stones, and A B. Does she win in the game which starts with one pile of A and one pile of B? (d) Hopefully there are some questions that naturally arose that you can pursue now. Can you determine who wins in other games of Halvsies? Who can guarantee a win in the original game set-up?

Answer rating: 100% (QA)

This is a combinatorial game theory problem involving strategy for a twoplayer game called Halvsies The questions here are about finding who has a win... View the full answer