Batch 2 - Class 276 - Dots and Boxes

• (Geoffrey - 64) Mr Houseman wishes to lay down carpet on the floor of his kitchen, which is exactly 12 feet x 12 feet. He has a piece of carpet 16 feet x 9 feet, which is just enough to cover the area. However, he doesn't want to cut it in any more pieces that absolutely required. What is the minimum number of pieces he needs to cut the carpet to fit it back on the kitchen floor?
• Answer: 2 pieces

• Explain the game of dots and crosses, and let kids play amongst themselves a few times on a 4 dots x 4 dots board
• Play with the kids (preferably as second player) and adopt the swastika pattern to try and keep the long chains to odd number. If kids ask you to play as first player, try to get side chains to get to even number of long chains, even though you might not be able to always do it
• See if kids notice what you are doing
• If not, ask them to observe
• Some games may get to double cross status - double crosses lead to optionality which allows the game to flip, so its best not to give that option to the opponent
• Is there a winning strategy for a 4x4 game? Who should win this board? How can that person force a winning position?
• How will you play if you were playing second?
• How will you play if you were playing first?
• Explain the concept to kids
• Why do long chains matter? Because they allow you to take one and then leave a double cross, thereby flipping the game at will.. In effect, they allow you to take what you want and leave the other person in a position you want to leave them to
• What will happen on larger boards? Note that double crosses imply optionality, which allows a flipping of position. So the first player should really try to get to even number of long chains + double crosses
• Larger boards get computationally intensive because its hard to see how many long chains are likely to form
• Try another size board, say 3 squares x 5 squares (notice that is 4 dots x 6 dots) - Again second player should win this game

• Answer: The approach is to calculate the volume of gulab jamuns in each bowl. It turns out that the second bowl has 20% more volume than the first, but is priced 25% higher. So the shop keeper actually makes more profit per bowl. Customer of course are happy to get gulab jamuns at lower price per piece.
• Instructor Notes: Since kids may not know trigonometry, let them come up with the approach, and then one may suggest that the volume of the smaller gulab jamun is about 80% of the volume of the larger ones