Batch 2 - Class 274 - Devil and Daniel Webster; others
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Preclass Exercise:
What if multiple people are infected? Say exactly two people? How would you solve the above problems? What if exactly 3 people were infected? What if an unknown number of people were infected?
Note that in the worst case, you may have to land up testing everyone, but you can perhaps save probabilistically. If the density increases a lot, then you are better off just testing everyone individually - the tree approach there would lead to 2^n tests.
We had fewer tests than people to be tested, and hence we looked at pooled testing
We didn't want to do this sequentially, so we looked at pooled testing in parallel. For 4 people, it took us two tests, for 8 people three tests and so on
Illustrate this to students as points on 2D square, 3D cube, and so on. Outline which people each test was testing, and the overlap to determine who is infected
Relate to information coding - we need similar number of bits to encode a given set of symbols.
We had a clumsy technician who would drop one test, so we introduced some redundancy
Explain notion of "distance" in two binary numbers, and why the distance of 2 assures us results here (with reference to 2D square, 3D cube and so on)
This is also the same as in encoding, we would correct for one error - show the movement along edges of 3D cube when an error happens
Note that we can not correct an error with this. We need a distance of 3 to do that
Another way to think about correcting for one error is to introduce a parity bit. Draw to illustrate the movement along edges. Both these translate into one extra bit.
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 Devil
Should Daniel work if there were a 1% cut for the Devil?
Homework Problem:
Mathematics in Pizzeria
In a pizza outlet, they are offering a medium and a small pizza for the same price as a large pizza (the quality, thickness etc are all the same). You do not have a ruler to measure lengths, but happen to have a protractor. How would you decide whether to buy small+medium or the large pizza?