(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.
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
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
(BerkeleyMC Chap 2 - Prob 10) How many even three digit numbers are there with no repeating digits?
328
Instructor Notes: First take through all 3 digit numbers (900). Then 3 digit numbers with no repeating digits (9.9.8). Hint on the problem (1) Lets start with rightmost digit instead of leftmost. (2) Lets separate the cases where 0 is the rightmost digit versus not. Make sure kids understand the addition of two cases. (BerkeleyMC has a good analysis)
(General Puzzle) Two geniuses are each assigned a positive integer and are told that the two numbers differ by 1. They then take turns to ask each other, ‘Do you know my number now?’. If the geniuses always respond to questions truthfully, prove that one of them will eventually answer affirmatively.
Principal 3: Counting the complement requires subtraction
(BerkeleyMC Chap 2 - Prob 12) Three different flavors of pie are available, and seven children are given a slice of pie so that at least two children get different flavors. How many ways can this be done?
Answer: 3.3.3.3.3.3.3 - 3 = 2184
Instructor Notes: Explain the problem clearly, for example aaaabbc is one possibility. Then calculate number of possibilities without a constraint (3.3.3.3.3.3.3). Then exclude cases. Make sure kids understand why subtraction happens.
(MC Chap 2- Prob 46) We toss the dice three times. How many possible outcomes with at least one occurrence of 6
(6.6.6 - 5.5.5)
Instructor Notes: Ensure kids understand the subtraction
Principal 4: n distinct items can be arranged in n! ways
Instructor Notes: Have kids work with 2, 3, 4 coins of different colors to figure out number of possibilities of arranging them. Drive the insight that number of ways with 4 coins is 4 times the number of ways with 3 coins. Hence introduce the notion and symbol for factorial. Introduce terminology of permutations. Get them to compute factorials for different numbers. Show them how fast it grows.
(Shakuntala - 219) What is the number of batting orders for 9 batsmen?
Answer: 9!
Homework Problem
Count number of zeros in 320!
Answer: 78. Each multiple of 5 gives a zero (there are enough even numbers available). Multiples of 25 have two 5 factors, and those of 125 have 3. So total 64 for multiples of 5, another 12 for those that are multiples of 25 as well, and another 2 for multiples of 125.
References:
Mathematical Circles (Russian Experience), by Dmitri Fomin, Sergey Genkin, Ilia Itenberg
More Puzzles, by Shakuntala Devi
A Decade of the Berkeley Math Circle. The American Experience, Volume 1. Zvezdelina Stankova, Tom Rike