1. Feb. 2005Counting ProblemsSlide 1 Counting Problems A Lesson in the “Math + Fun!” Series

2. Feb. 2005Counting ProblemsSlide 2 About This Presentation EditionReleasedRevised FirstFeb. 2005 This presentation is part of the “Math + Fun!” series devised by Behrooz Parhami, Professor of Computer Engineering at University of California, Santa Barbara. It was first prepared for special lessons in mathematics at Goleta Family School during the 2003-04 and 2004-05 school years. The slides can be used freely in teaching and in other educational settings. Unauthorized uses are strictly prohibited. © Behrooz Parhami

3. Feb. 2005Counting ProblemsSlide 3 Counting Is Easy, Isn’t It? True, when the number of things is small or the items to count are neatly laid out But it can be tricky or hard How many tiles? How many animals are there in this picture?

4. Feb. 2005Counting ProblemsSlide 4 How Not to Count “A cow has 12 legs, 2 in front, 2 in back, 2 on each side, and 1 in each corner.” – N.J. Rose How many legs does a cow have?

5. Feb. 2005Counting ProblemsSlide 5 Remove exactly four dots so that no square can be formed with the dots that remain. 5 + 4 + 2 = 11 Counting Squares: Easy Version How many squares can you find that have four of these dots in the corners? Other answers?

6. Feb. 2005Counting ProblemsSlide 6 Counting Squares: Harder Version How many squares can you find that have dots in their corners? Remove exactly six dots so that no square can be formed with the dots that remain. 9 + 4 + 2 + 4 + 2 = 21

7. Feb. 2005Counting ProblemsSlide 7 How Many Squares Do You See? 4  1  4 + 4 + 4 + 1 + 1 = 14 squares

8. Feb. 2005Counting ProblemsSlide 8 Hint: Count by side colors: 3 sides green (solid), 3 sides blue (dotted), 3 sides red (dashed), 2 sides green 1 side red, 2 sides green 1 side blue,... Activity 1: How Many Triangles? 12

9. Feb. 2005Counting ProblemsSlide 9 In how many different ways can you arrange the letters O, P, S, T? How many of these arrangements form ordinary English words? OPST OPTS OSPT OSTP OTPS OTSP Activity 2: Counting Arrangements POST POTS _____ SOPT SOTP _____ In how many different ways can you arrange the letters O, P, T, T? In how many different ways can you arrange the letters O, T, T, T? The number of ways is: 4  3  2  1 Can you explain this rule and use it to find the number of arrangements when there are five distinct letters to arrange? Explain your answer like the previous puzzle and test it on five letters with one repetition (O, P, S, T, T) Explain your answer like the first puzzle and test it on five letters with 2 repetitions (O, P, T, T, T) OPTT OTPT OTTP _____

10. Feb. 2005Counting ProblemsSlide 10 How Many Ways to Have 25¢ in Change? and 10  and 15  5  4  and 5  3  and 10  2  and 15  and 5  1  and 20  25  13 different ways

11. Feb. 2005Counting ProblemsSlide 11 ___ different ways 5  10  15  5  4  5  and so on... Activity 3: How Many Ways for 50¢ in Change?

12. Feb. 2005Counting ProblemsSlide 12 Dividing Chocolate Bars Divide an 8 x 4 chocolate bar into individual pieces using the smallest number of breaks. How many breaks did you need? Interactive version at: http://www.cut-the-knot.org/ctk/memes/shtml After 1 break After 2 breaks After 3 breaks After 4 breaks

13. Feb. 2005Counting ProblemsSlide 13 A Couple of Smaller Examples Each break increases the number of pieces by 1. So, to go from 1 piece to 6 pieces, we need 6 – 1 = 5 breaks. 1 2 3 45 1 2 3 4 5 6 7

14. Feb. 2005Counting ProblemsSlide 14 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 How many times does each digit 0-9 appears if we write down all numbers from 1 to 20? Two 0s, 3s, 4s, 5s, 6s, 7s, 8s, 9s Activity 4: Count the Digits Three 2sTwelve 1s 1 2 3 4 5 6 7 8 9 10.......... 100 How many times does each digit 0-9 appears if we write down all numbers from 1 to 100? Answer the question without writing down all the numbers. How many 0s? ____ How many 1s? ____ How many 2s? ____ How many 3s? ____ How many 4s? ____ How many 5s? ____ How many 6s? ____ How many 7s? ____ How many 8s? ____ How many 9s? ____ Check your answers: Add all the numbers, and compare against 192. (Why 192?)

15. Feb. 2005Counting ProblemsSlide 15 How Many Handshakes? Three people can shake hands in three different ways. A & B; A & C; B & C A BC In how many different ways can four people shake hands? A B C D A & B C DB & C D C & DA & B C D EB & C D EC & D ED & E A B C D E What about five people? The numbers 1, 3, 6, 10,... are known as triangular numbers 6 10

16. Feb. 2005Counting ProblemsSlide 16 Triangular Numbers Number of balls in each tier, beginning from the top: 1, 4, 9, 16, 25,... Total number of balls for different numbers of tiers 1, 5, 14, 30, 55,... See if you can relate the numbers shown on the right to numbers in Pascal’s triangle above. Pascal’s triangle 11 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 16 15 20 15 6 1 1 7 21 35 35 21 7 1 11 2 +

17. Feb. 2005Counting ProblemsSlide 17 Routes on a Street Grid A Q R S W B How many different ways are there to get from point A to point B if we want to use a shortest path? 1 12 11 34 13610 1 B A 1 1 2 11 34 13 6 1 B A QRS TUV XYZ W A Q U Y Z B A T U V Z B A Q R V W B Pascal’s triangle! 11 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1

18. Feb. 2005Counting ProblemsSlide 18 Activity 5: How Many Ways to Trace a Word? M I I N N U U S N (1) Start from M on the left and go to the S on the right, spelling the word “MINUS.” In how many different ways can you do this? S Q Q U U A A R U A A R E E S R (2) Do the same with the word “SQUARES” below. M A A D D A A M D A A M A A D M M A D A M D A A M (3) Do the same with the word “MADAM,” beginning from any of the four corners and moving in any of the four directions. (4) Extra challenge: The same as (3), but you can start from any of the seven Ms.

19. Feb. 2005Counting ProblemsSlide 19 Counting by Estimation: Spectators in a Stadium

20. Feb. 2005Counting ProblemsSlide 20 Counting by Estimation: Cells Under a Microscope

21. Feb. 2005Counting ProblemsSlide 21 Tricky Counting: Count the Black Dots

22. Feb. 2005Counting ProblemsSlide 22 Next Lesson Thursday, March 3, 2005 A problem to think about: You have a 3-cup container and a 5-cup container only. How do you measure one cup of sugar? 3 5