Batch 2 - Class 95 - Exploring Zero, Leibniz' Triangle

• Create your own bike that move on a catenary. See http://www.exploratorium.edu/square_wheels/square_wheels.pdf 

• Play the following puzzle on an infinite triangular array (only the first six rows are shown here). The puzzle begins with three coins placed at the apex of the array and a move consists of removing a coin from the array that has two empty cells just below it and replacing it with two coins in those empty cells. For example, two moves to start the game are shown. 

• Let kids attempt for a while. Some kids may come to the conclusion that it seems to go on. How can we prove that this is not doable?
• Introduce Leibniz' Harmonic Triangle. Unlike Pascal triangle, where any circle has a number which is sum of two circles above it, Leibniz' Triangle circle has a number which is sum of two circles below it. Ask kids to construct a sample Leibniz' Triangle.
• If they have not got to the 1,1/2,1/4 Leibniz triangle, help them get there. Now do you see a solution?
• What is the invariant here when we make a move?
• At any stage of the game, what should be the total of all covered cells?
• What is the possible total besides the top three cells? How many moves does it take to create a total equal to top three cells?

  Answer: In any finite number of moves, it is not possible to get to total of 2 without having one of the top three cells covered.
• Can it even be done in infinite moves? What are the kind of patterns that develop as we "roll out" the coins? How does the triangle fill up at the edges relative to the center?

  Introduce the "real" Leibniz' Harmonic Triangle, which is reciprocals of Pascal triangle multiplied by 1/n for nth row. Notice that this has the same property and hence our coin game will work on this triangle as well.But, what is the sum of left edge here (1/1+1/2+1/3+1/4 ... = infinite), which means that if we desire a sum of two then we can't go on infinitely on edges, so we need to go "more than infinite" in the center!?!? In fact 1/3+1/4+1/5...1/19 > 2.

• Everyone's heard of odd-even rule? Some people whose number plates were ending with zero chose to drive on both odd and even days. Were they right? Is zero odd or even? Can you reason it?

  How do we define odd and even numbers? If we can split a pile into two piles of equal sizes, then it is even. Does zero behave that way? Well, if we decide that zero is countable...Mathematically, if a number b can be written as 2a, where a is a counting number, then b is even. Does that hold for zero? Well, if we decide that zero is countable...
• Why can't we divide by zero? What is 5/0? Is 0/0 ok?

  Why is 20/4=5? Because 20=4*5. What happens with 5/0 - can we find a number so that 5=0*a?Why is 0/0 a problem then? Well, every number is an answer, so it is again undefined!Why do we call 5/0 as infinite? Is limit same as the exact answer?
• Is -0 same as 0? Why/ why not?

  -a is a number which when added to a gives a total of 0. Is 0-0=0?But 0+0=0, so -0 should be same as 0?
• Does 0 have a square root? What is square root?

  a is square root of b if a*a = b. This is the mathematical interpretation. Geometrically, a is side of square whose area is b... What does square root of negative numbers mean?What does square root of 0 mean in geometric and arithmetic terms?
• Why is 0! equal to 1?

  What does factorial mean - product of all numbers from 1 to the number... So what is 0!?However, if we see nCk as ways to select k people out of n, then there is only one way to select 0 people out of any number of people (sit back and do nothing). So what is 0!? As matter of convenience, we assign it a value of 1.So what is (-3)!?
• Why is 2^0 =1?

  2 multiplied by itself 0 times doesnt seem to have any meaning?If we insist that 2^a*2^b=2^(a+b), then what can we get to? Note that this rule does not have to be true for non-countable numbers...If so, what is 2^(-1)?So again, 2^0=1 largely for arbitrary reasons!
• What is 0^0?

  If we take 1^0, 0.1^0, 0.01^0 and so on, the limit of 0^0 approaches 1If we take 0^1, 0^0.10, 0^0.001 and so on, the limit of 0^0 approaches 0Which is it then? How about 36? Since 0^0*0^3=0^3, and since 0^3 is 0, 0^0 can be 36 (or any other number)Since it seems to have all solutions, it is undefined

• (Moscow - 174) Wheels A,B,C and D are connected with belts as shown. If wheel A starts to rotate clockwise, can all the 4 wheels rotate? In which direction would each wheel rotate?

• Can the wheels rotate if all 4 belts are crossed?
• What is belt 1 and 3 are crossed?