Batch 3 - Class 234 - Bubbling Cauldrons

• Simpson Paradox: When one analyzes the voting records of democrats and republicans during the American Civil Rights Act, 1964, following facts emerge
• In the northern states, 94% of Democrats voted for the act, compared with 85% of Republicans
• In the southern states, 7% of the Democrats voted for the act, compared to 0% of Republicans
• Who showed more support for the act - Democrats or Republicans, as percentage of total Democrat and Republican population respectively.
• Answer: Consider the following:

• What causes this non-intuitive answer? While there is abysmal voting by republicans in south, there are only 10 of them, so even if one vote had swung, it would look like 10%. On the other hand, there are lot of democrats in South, but very low support.

• Start by having two cauldrons and frogs/ numbers being dropped into it. Let students drop the numbers into cauldrons, and at a certain point the cauldron blows up. Don't tell the rule (a cauldron blows up when the number being added is a sum of two prior numbers in that cauldron) and let students figure out the rule through observation
• Now, challenge the students to drop the numbers so that they can get to maximum number of frogs in cauldrons. It may take a bit of hit and trial. Optimal answer is as under:

• Now pose the three cauldron problem. Let kids play once collaboratively. 
• Divide the kids into teams to devise their strategies - let them think it over for 5 minutes. Assign names to teams, and lets play
• Optimal solution for 3 cauldrons

• Issai Schur posed this problem of integer partitioning 1916. There is an unsolved problem here - things get very difficult with 4 pots. No one knows solution for 5 pots or higher.
• Extension: In the two cauldron problem, what if the first frog escapes? How many frogs starting the second frog can you put in cauldrons? What if first two frogs escape? and so on?
• The solution to above is believed to be 2-12, 3-17, 4-22 and so on n to 5n+2

• (Dudeney 304) Arrange the Ace, King, Queen and Jack of all four suits in a 4x4 square so that none of the rows, columns of main diagonals have the same value or same suit occurring twice.