Batch 2 - Class 03 - Binary Counting

• On a 8x8 chess board, there is a lame tower, that players can successively move either any number of steps to the left, or any number of steps down. Figure out a winning strategy. When can you win as the first player.

  Answer: The first player wins on all other positions except the diagonal, where the second person wins. The winning strategy for any player is to get the tower to a diagonal position
• Ternary Notation: Write 1-30 in base 3 notation. Reinforce the rules. 
• 9 balls, one is lighter. Find which in two weighings (Answer: Weigh 3 balls against three others. Light/Heavier/Same outcomes point to the three ball group which has the defective ball. Then within that group, weight two balls against each other for another light/heavier/same outcome which tells which ball it is)

• Ternary system: http://www.geometer.org/mathcircles/CardTricks.pdf - Problem 1, Base 3 math through card trick - 27 cards in three columns, everytime the subject tells which column the card is in. You assemble that column in middle of other two columns. At end of three assemblies, chosen card is the center card.
• Bill Gates' Bankruptcy Puzzle - Bill Gates wanted to enroll himself for our math circle class. I told him I normally charge kids Rs 5000 for the class, and for him I would charge $200, and indicated that we may have 10 classes for that. Bill Gates came back with a request for a discount. So I told him, fine, you pay me $1 for the first class, $2 for the next, $4 for the next and so on. He readily agreed. He came back after 3 months to pay the amount, and wasn't happy about the discount anymore. Why? What would happen if we continued this arrangement for a year? ($1T - more than 12 times of Bill Gates' worth of $80B) In one more quarter, I would own more than total world's wealth ($200 Trillion).

• Labyrinth - Pg 83: Test of Ingenuity) Sherlock Holmes puzzle - In a room, there are two lengths of strings hanging, each 6 ft long. The height of the room is 10 ft. The strings are 10 ft apart. You are given a swiss army knife, a firecracker, a small vial of ether, a 10 kg block of ice, and a kitten. The problem is to tie the two ends of the string together (you can't hang on the string!)

  Ask kids to doodle binary trees (Khan Academy Videos) to get a first hand feel or how fast this grows. Ask for volunteer first on basis of time bids, and then let them realize the magnitude of underestimation. Crashing binary trees -> Fractals. https://www.khanacademy.org/math/recreational-math/vi-hart/doodling-in-math/v/doodling-in-math-class--binary-trees 
• King with 1000 bottles of wine. One gets poisoned. 10 disposable servants. 3 weeks for poison to take effect, 4 to a party. How does the king find out the safe 999 bottles. http://alokgoyal1971.com/2014/07/06/puzzle-66-all-the-kings-wine/

  Answer at http://alokgoyal1971.com/2014/07/12/solution-to-puzzle-66-all-the-kings-wine/ 

• Place Eight queens on a chessboard, so that none of them can capture each other.
• (MC Chap 0, Prob 1) - There is a jar of yeast and the yeast doubles every day. It fills the jar on 30th day. When was the jar half full?

  Answer - Day 29