Batch 3 - Class 112 - Dots and Boxes
Pre-Class Problem:
- How can you divide a long paper strip into 5 equal parts?
- Answer: start with an approximation from one end. Then fold the far end into four parts. The new 1/5th estimation is better than the old one. Repeat few times to get a good approximation of 1/5
Attendance: Anisha, Tishyaa, Nandini, Aneyaa, Khushi, Arnav, Anishka, Liza, Ishaan, Kabir, Diya, Arnav, Gautam
Class Notes:
Dots and Boxes
- 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
Gulab Jamuns
An school canteen sells two gulab jamuns for Rs 12 in a bowl, such that the perfectly spherical gulab jamuns fill the bowl. A shop outside the school introduces a new offering to sell three gulab jamuns for Rs 15 in a bowl of same size, so that the perfectly spherical gulab jamuns fill the bowl. Kids are excited at getting gulab jamuns at Rs 5 per piece rather than Rs 6 per piece! But is this a good strategy on part of the new shop?
- 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
Homework
- Six other people were in a mansion when Mr. Edwards was murdered in the attic, but each has an alibi backed up by another person. Ms. Abbott says that she saw Vicar Carter reading in the study at the time. Mrs. Blyth says she saw Ms. Doyle exit the mansion for a walk. Vicar Carter says he saw Mr. Gibson walking the grounds as he looked out the window. Ms. Doyle says she saw Mrs. Foseberry in the garden. Mr. Gibson says he saw Ms. Abbott dusting the hallway. Mrs. Foseberry says she saw Mrs. Blyth in the sitting room through the window. Each person confirms that they were doing what they were observed to be doing. If at least one of the six were involved in the murder, what is the minimum number of them who are lying?
- Answer: Two
- Reference: https://www.expii.com/solve/41/1
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