Batch 3 - Class 125 - Cake Pies (Combinatorics)
Pre-Class Problem:
(Dudeney - 284) In the illustration we have a somewhat curious target designed by an eccentric sharpshooter. His idea was that in order to score you must hit four circles in as many shots so that those four shots shall form a square. It will be seen by the results recorded on the target that two attempts have been successful. The first man hit the four circles at the top of the cross, and thus formed his square. The second man intended to hit the four in the bottom arm, but his second shot, on the left, went too high. This compelled him to complete his four in a different way than he intended. It will thus be seen that though it is immaterial which circle you hit at the first shot, the second shot may commit you to a definite procedure if you are to get your square. Now, the puzzle is to say in just how many different ways it is possible to form a square on the target with four shots.
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- Answer: Twenty-one different squares may be selected. Of these nine will be of the size shown by the four A's in the diagram, four of the size shown by the B's, four of the size shown by the C's, two of the size shown by the D's, and two of the size indicated by the upper single A, the upper single E, the lower single C, and the EB. It is an interesting fact that you cannot form any one of these twenty-one squares without using at least one of the six circles marked E.
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Attendance: Muskaan, Damini, Siddhant, Tishyaa, Liza, Ahana, Arnav, Khushi, Palak, Arnav, Anishka
Class Notes:
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 = 24
- Prob 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)
- Instructor Notes: This is the overall problem, which can be taken in multiple stepsFirst step is for kids to realize that there are rectangles of different sizes. They might start to count all, which is fine.Lets break down to smaller problems first. Can we do this for a 2x2 chessboard? 3x3? 4x4? s there a pattern?What is an even simpler problem - lets see how many line segments we can have in a line of length 2. Length 3? Length 4? What is the pattern?Is there a linkage between the line pattern and chessboard pattern? Why?Can we now arrive at a general way to find number of rectangles in a large chess board - 8x8?
- 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
- Cake Pies: Consider a cake which has 8 sections and 3 layers. In how many ways can you cut a pie taking any number of adjoining sections and any number of adjoining layers?
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- Answer: 342. The number of layers' combinations is 6, like in the previous problem. For the sections, there are 8 ways to take a single section, 8 ways to take 2 adjoining sections, 8 ways to take 3 adjoining sections, and so on. Except that there is only one way to take all the eight sections - so total 57 ways to choose the sections. 57x6=342.
- (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: 64.49
- What 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
A Decade of the Berkeley Math Circle. The American Experience, Volume 1. Zvezdelina Stankova, Tom Rike