[Image courtesy of SharpBrains.com.]
A week ago, we shared a brain teaser sent in by a PuzzleNationer named Bethany, who challenged us to solve the following stumper.
Today, we’re going to share not only the solution, but how we got there! Please enjoy this brief solve and tutorial, inspired by one of your fellow PuzzleNationers!
The Peppermint Patty Riddle
You’re facing your friend, Caryn, in a “candy-off,” which works as follows: There’s a pile of one hundred caramels and one peppermint patty. You and Caryn will go back and forth taking at least one and no more than five caramels from the candy pile in each turn. The person who removes the last caramel will also get the peppermint patty. And you love peppermint patties.
Suppose Caryn lets you decide who goes first. Who should you choose in order to make sure you win the peppermint patty?
When I posted this brain teaser, I said that “the question basically demands that you not only achieve victory, but figure out how to do so with your very first move.”
That was meant to be a hint. Because the best way to unravel this riddle is to start at the end.
Look at the last turn. If there are five or fewer caramels left on the board and it’s Caryn’s turn, she wins. For you to guarantee a win, she needs to see six caramels on her last turn. (Seven or more allows her to take just enough to leave YOU with six, which means she wins on her next turn.)
And that six is the magic number. Her previous turn, you need to leave her with 12 caramels. The turn before that, 18 caramels. The turn before that, 24. And so on and so on.
Let’s look at the 12 caramel turn. If she takes any number of caramels (anywhere from 1 to 5), she’ll leave you with between 11 and 7 caramels. You can use your turn to remove whatever number of caramels are necessary (anywhere from 5 to 1) to leave her with six.
The entire game can be broken into six-caramel increments.
And the closest multiple of 6 to 100 (your starting point) is 96 caramels.
Which means you can’t let Caryn go first. You need to remove 4 caramels on your first turn, and then use every subsequent turn to ensure Caryn faces a multiple of six, which eventually gives you the victory and the peppermint patty.
So the answer to the riddle is you should go first.
Did you solve the brain teaser, fellow PuzzleNationer? Or do you have a brain teaser you’d like us to unravel with you? Let us know in the comments section below! We’d love to hear from you.
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