Batch 2 - Class 82 - Devil and Daniel Webster

Pre-Class Problem:

Leaderboard challenge on https://www.expii.com/ramanujan
Anishka and Muskaan claimed their Level 1 prizes!

Attendance: Smiti, Muskaan, Tishyaa, Diya, Arnav, Anishka, Nandini

Class Notes:

The Devil and Daniel Webster is a short story by Stephen Vincent Benet about a New Hampshire farmer who sells his soul to the Devil and is defended by American statesman Daniel Webster. The problem below uses this context, which provides an interdisciplinary connection, although the problem given here is not taken directly from the story.

Set up the problem by reading the following to students:

The devil made a proposition to Daniel Webster. The devil proposed to pay Daniel for services in the following way:

On the first day, I will pay you $1000 early in the morning. At the end of the first day, you must pay me a commission of $100; so, your net salary that day is $900. At the start of the second day, I will double your amount left at the end of the previous day. So, at the beginning of the second day, I will pay you $1800; but at the end of the second day, you must double the amount that you pay me to $200. Will you work for me for a month?

Ask students to describe the pattern for Day 3, Day 4, and so on. Students should recognize that Daniel's salary at the beginning of Day 3 will be $3200 (since the amount he had at the end of Day 2 was $1600), and he will need to pay the Devil an amount of $400 at the end of Day 3 (since he had paid $200 on the previous day).
Does the salary scheme work? Why/ Why not?
How many days would you work for the devil?
Can you try and plot the curves that result for Daniel's pay and Devil's commission?
Can you express the Devil's commission and Daniel's money at end of each day as a formula?
Randomize to other set of values for initial pay, commission and factor for each student
See if some of these lead to viable salary schemes
What if the devil asked for 80% commission, but no doubling?
How long should Daniel work for the devil? For three weeks? For a month?At what rate is it most advantageous to both Daniel and DevilShould Daniel work if there were a 1% cut for the Devil?

Probability Paradox! (Expii.com)

You work for the census bureau. You call a random phone number and learn that the household has exactly two children and at least one of them is a girl. What is the probability that they are both girls?
Answer: 1/3
The next day at the census bureau, you call another random phone number. This household also has exactly two children, and at least one of them is a girl who happened to be born on a Tuesday. What is the probability that they are both girls?
Answer: 48%
What do you think is going on? Why did we think of 1/2 as an intuitive answer for the first problem, and then it turned out to be 1/3? Why did it get closer to 1/2 in second case? What do you think will happen if we realized that at least one of the kids is a girl who happened to be born at 12:30pm on Tuesday?

Homework

A chess queen can move any number squares vertically, horizontally, or diagonally. A Queen On Roller Skates (QORS) must keep moving until she encounters a wall or other obstruction. Scott Purdy asks the question "What is the minimal number of obstructions that allow a QORS to stop on any square?"
For 8x8 board For 9x9 boardFor 10x10 board
Answers: 8 by 8: six blocks:
OOOOOOOO
OOOOOOOO
OOXOOXOO
OOOXOOOO
OOOOXOOO
OOXOOXOO
OOOOOOOO
OOOOOOOO
9 by 9: nine blocks:
OOOOOOOOO
OOOOOOOOO
OOXXOOXOO
OOOOOOXOO
OOOOXOOOO
OOXOOOOOO
OOXOOXXOO
OOOOOOOOO
OOOOOOOOO
10 by 10: twelve blocks:
OOOOOOOOOO
OOOOOXOOOO
OOXOOOOXOO
OOOOOOXOOO
OOOOOXOOXO
OXOOXOOOOO
OOOXOOOOOO
OOXOOOOXOO
OOOOXOOOOO
OOOOOOOOOO

References:

http://illuminations.nctm.org/Lesson.aspx?id=1135

http://expii.com/ramanujan

http://www.mathpuzzle.com/Solution.htm