You’re with your best mate Bella in a battle for the best chocolate bars. On this unlikely occasion, your favourite is the Bounty bar surrounded by a sea of Mars bars.
To be precise, there are 100 Mars and just one Bounty. You and Bella will take it in turns to take 1 to 5 Mars from the pile each turn. Whoever removes the very last Mars will then claim the Bounty that you both crave. You really want to win that Bounty!
Bella lets you decide who goes first. Do you go first, or let her go first, to ensure you claim the Bounty?
Start at the end. If Bella is faced by 1 to 5 Mars on her turn, she will take them all and win the Bounty. So, you need to make sure this situation doesn’t occur. If there are 6 Mars, then even if she takes 1, you’ll take 5 and win. But, if she starts with 7 to 11 Mars, she can take just enough Mars to leave you with too many to claim the Bounty.
6 is the ideal target; you can achieve this if she has 12 Mars on the turn before that. In that case, she’ll leave you with a minimum of 7 Mars, so you can just take one, leaving her with 6, thus claiming the Bounty.
Then, you can make sure she has 12 by ensuring she has 18 on the previous turn. To leave her with 18, you’ll need 24. If you leave Bella with multiples of 6, she’ll never win.
100 isn’t a multiple of 6 – but 96 is. If you go first, you take 4, leaving Bella with 96. Then, you can leave her with 90, 84, 78, etc, in multiples of 6, to ensure you leave her with just 6 at the end. She’s forced then to take a single Mars – so you will win the Bounty.
