Batch 2 - Class 06 - Combinatorics (1)

Homework Puzzles (Split discussion on these to beginning to each section)

[Khan Academy, Liar Truthteller brain teaser] - 2 doors, one has treasure, one has hell. Two door keepers, one always tells truth, the other always lies. One question to one doorkeeper to decide which door to open.
Answer: "what would the other guy say which is the correct door to open"
(Contributed by Tara) Truthtellers and liars. There is a village of truthtellers (who always tell the truth) and liars (who always lie). I want to go to the village and reach a fork in the road, where a villager is standing. Can I ask one question to the villager, without knowing whether she is a truthteller or a liar, which will allow me to determine which path to take on the fork to reach the village
Answer 1: "If I asked you which road goes to the village, what would you say?" (Take the road the villager points to. Logic is Truth->Truth = Truth, and Lie->Lie = Truth)Answer 2: "If I asked a member of your opposite tribe the road to the village, what would they say?" (Take the road the villager does not point to. Logic is Truth->Lie = Lie, and Lie->Truth = Lie)
[MartinShCol 5.22]: Four bugs at corner of squares start moving directly towards each other. Trace their path. How long does each bug travel before they meet
Answer: Spiral converging at centre. Distance is side of square

Attendance Smiti, Tishyaa, Armaan, Hriday, Aastha, Avy, Kushaan, Tara, Adit, Anisha

Class puzzles

Combinatorics

Note: The symbol "." is used as a multiplication sign below

Principal 1: If the thing we are counting is an outcome of a multistage process, then the number of outcomes is the product of the number of choices for each stage

(MC Chap 2, Prob 3) Three towns A, B, C in Wonderland. Six roads go from A to B and four roads go from B to C. In how many ways can one drive from A to C?
Answer: 6.4 = 24Prob 4 - Add town D, with 3 paths from A to D, and 2 from C to D - now how many paths? (Answer: 24+3.2=30)
(MC Chap 2, Prob 6) A natural number is odd-looking if all its digits are odd. How many odd looking 4 digit numbers are there?
Answer: 5.5.5.5 = 625
(MartinShCol - 1.1) A date is ambiguous if it can be validly written both as DD/MM/YYYY and MM/DD/YYYYY (for example, 08/09/2001 is ambiguous because it can mean 8th Sep or 9th Aug). How many ambiguous dates are there in a year?
Answer: 12*11 = 132 (Note that 03/03/2001 is not ambiguous so every month has 11 ambiguous dates)

Principal 2: If the thing we are counting can happen in different exclusive ways, then the number of outcomes is the sum of the number of outcomes through each way

Instructor Notes: Key is for students to understand when to add and when to multiply.

(Shakuntala - 229) How many rectangles does a chessboard have (1296)
Answer: 1296One way is to try and count rectangles of each size and add them upAn easier way is to see that each horizontal line segment along a side of the chessboard (of different sizes 1, 2, 3...8) and each vertical line segment along the perpendicular side produce a rectangle. Hence number of rectangles is square of the number of line segments. Number of line segments along a side are 8 + 7+ 6+... +1 = 36. The number of rectangles is 36.36 = 1296

Instructor Note: First step is for kids to realize that there are rectangles of different sizes. Then they will start counting all, which is fine. Introduce the notion of how a rectangle is formed by two line segments perpendicular to each other

(Manas contributed this puzzle) Put mathematical signs between four 9's to make 100
Answer: 99+9/9 , 99/.99 (Here "." is a decimal!)
(MC Chap2, Prob 13) In how many ways can you place a white and a black rook on a chessboard so that they can't capture each other?
Answer: 64.49What if both rooks are identical? Answer: 64.49/2
Homework Problem (MC Chap2, Prob 14) In how many ways can you place a white and a black king on a chessboard so that they can't capture each other?
Answer: 3612 (4.60+24.58+36.55)

Instructor Notes: First insight is that the number of squares attacked depends on the position of the king. Second, kids should correctly "add" the different scenarios.

References:

Mathematical Circles (Russian Experience), by Dmitri Fomin, Sergey Genkin, Ilia Itenberg

The Colossal Book of Short Puzzles and Problems, by Martin Gardner

More Puzzles, by Shakuntala Devi

https://www.khanacademy.org/math/recreational-math/puzzles/brain-teasers/v/liar-truthteller-brain-teaser