Batch 2 - Class 41 - Game Theory - Casino Games

• 20 digit number - Two players take turn writing a 20 digit number left to right. If the number is divisible by 3, Bob wins. Else Alice. When does who win? (Answer: If Bob starts, Bob loses. If Alice starts, Bob Wins)

  What if the number above has to be divisible by 15? (Answer: Bob always loses)

• Triangle problem: arrange the numbers 1 through 6 into a “difference triangle”, where each number in the row below is the difference of the two numbers above it. For example, 6 4 1, 2 3, 1 almost works, but it has two 1s and no 5. Variations: What about with 10 numbers? 15? What is the solution with the smallest range in the top row (largest minus smallest number)?

• Show kids a roulette wheel, and explain the odds (35:1, i.e. if you bet one and win, you get back 1 plus 35)
• Ask kids to figure out their chances of winning (5.26% house advantage)
• Consider other types of bets - red versus black (1:1 odds). What are the chances of winning
• Strategies to beat - ask students what they could do

  Is there a winning strategy if you have unlimited money? (Martingale strategy)Casinos allow fixed limit tables, and will often spot such bettingPractically, one will run out of moneyPredicting the quadrant where the ball will fall (Thorp and Shannon's wearable computer)Studying bias in wheel
• See real life Roulette exploits at https://en.wikipedia.org/?title=Roulette

• A CALENDAR PUZZLE. If the end of the world should come on the first day of a new century, can you say what are the chances that it will happen on a Sunday? 

  Answer:  The first day of a century can never fall on a Sunday; nor on a Wednesday or a Friday.

• MAMMA'S AGE. Tommy: "How old are you, mamma?" Mamma: "Let me think, Tommy. Well, our three ages add up to exactly seventy years." Tommy: "That's a lot, isn't it? And how old are you, papa?" Papa: "Just six times as old as you, my son." Tommy: "Shall I ever be half as old as you, papa?" Papa: "Yes, Tommy; and when that happens our three ages will add up to exactly twice as much as to-day." Tommy: "And supposing I was born before you, papa; and supposing mamma had forgot all about it, and hadn't been at home when I came; and supposing——" Mamma: "Supposing, Tommy, we talk about bed. Come along, darling. You'll have a headache." Now, if Tommy had been some years older he might have calculated the exact ages of his parents from the information they had given him. Can you find out the exact age of mamma? Note that ages might not be whole years, but some years and months.