Find the missing number in the sequence 10000, ?, 100, 31, 24, 22, 20, 17, 16, 15, 14, 13, 12, 11, 10
Answer: This is number 16 written in base 2 to 16. The missing number is 16 written in base 3, i.e. 121
AttendanceArjun, Muskaan, Arnav, Anishka
Class puzzles
Rope around the world puzzle (Credit: William Whiston 1667-1752)
Take a tennis ball, and choose one of its equators. How much longer would you have to make the rope so that it is one foot from the surface of the tennis ball at all points?
What if we took the equator of earth. How much longer would we have to make the rope so that it is one foot from the surface of the earth at all points?
Let kids guess the answers to each.
Answer: In both cases the answer is about 6.28 ft (2 pi) - a surprising conclusion given that most people expect the earth answer to be much higher.
Ultimate Tic-Tac-Toe: Lay reference of standard tic-tac-toe and its deterministic nature. Let kids play a couple of games.
Introduce the ultimate tic-tac-toe, with following rules:
Draw smaller tic-tac-toes in each square of larger tic-tac-toe
Mark a smaller square on each turn
When you get three in a row on a small board you win that small square
To win the game, win three small boards in a row
Next small board you play on is determined by which entry within previous smaller board your opponent played
If you are directed to a small board which is already full, then you get to choose which board to play on
If a small board is a tie, it neither counts as X nor O
Let kids play a few games.
Who wins the ultmate tic-tac-toe game? Is there a deterministic strategy?
A casino offers a game of chance for a single player in which a fair coin is tossed at each stage. The initial stake starts at 2 dollars and is doubled every time heads appears. The first time tails appears, the game ends and the player wins whatever is in the pot. Thus the player wins 2 dollars if tails appears on the first toss, 4 dollars if heads appears on the first toss and tails on the second, 8 dollars if heads appears on the first two tosses and tails on the third, and so on. Mathematically, the player wins 2k dollars, where k equals number of tosses (k must be a whole number and greater than zero). What would be a fair price to pay the casino for entering the game?
Answer: This game has an infinite expected value, and hence it is worth paying any finite price
However, most people have a trouble dealing with this. The reason is that people tend to discount extremely rare events -could follow up with black swan.Also, for any reasonable bank assets, the payout is not that large. For example, if the bank had World GDP of $80 trillion as its resources (and could not cover the bet for more than that,) then the expected value of the lottery is under $50.
Homework Problem:
Write a program to simulate the above game of chance. Run a billion (or more!) games, and find the average payout. If you run enough games, you should be able to see that there are some of the outlier wins, that make it worth a large bet.