Isn't \(100000\) such a huge number (Part-9)?

Discrete Mathematics Level 5

AG and SG both study in the same class. They are mad in love of Katrina, one of their prettiest classmates. Katrina also is fond of their intelligence and smartness.

So, on the upcoming Valentine's Day, they plan to have a date with Katrina. They fight with each other to go with Katrina on a date. Katrina listens to their fight and tries to calm down the matter.

Katrina: Hey, let's not fight over it!

SG: But I want to go with you on a date.

AG: I also want to go.

Katrina: I have an idea. Let's play a game. I have a pile of \(100000\) coins and I will give you any number of coins ranging from \(1\) to \(100000\). You can take out \(1\) coin, \(2\) coins or half of the pile of coins. You have to take turns to take out the coins. The one who takes out the last coin wins the game. Since I am a huge fan of your intelligence, I will go with the one who wins the game. SG will make the first move. So, are you ready guys?

AG: Sounds interesting!

SG: Let's play AG!!

Katrina will randomly choose how many coins to give to AG and SG. If both the players play optimally and intelligently, let the probability of AG to go on a date with Katrina is \(\dfrac{a}{b}\) where \(a\) and \(b\) are coprime positive integers. Find \(a+b\).

Details and Assumptions:

  • If the number of coins chosen by Katrina is odd, then the players can't take out half of the pile of coins. They can take out only \(1\) or \(2\) coins.

  • Bonus points for identifying AG and SG.

  • \(\color{Red}{\huge{Happy}}\) \(\color{Red}{\huge{Valentine's}}\) \(\color{Red}{\huge{Day}}\).

Image credit: Cartoon released by Archie Comic Publications Inc. in 2002.

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