1. Homework 3 Can you divide 36 balls into 9 groups such that each group has odd number of balls? 36 ÷ 9 = 4, 4 is even What if we change things around a little bit? 5, 3, 5, 3, … etc. Would it work? Note that 9 odd numbers added together must be odd. Impossible. 1

2. Homework 4 are 9 squares in a grid that each has a coin in it as below. Can you remove 4 coins such that each row and each column has odd number of coins? 2

3. Homework 4 are 9 squares in a grid that each has a coin in it as below. Can you remove 3 coins such that each row and each column has even number of coins? 3

4. Olympiad Math III Lesson 10 Area of patterns on grids 4

5. Purpose Calculating area of patterns on a grid improves your understanding of the patterns Prepare you for formal geometry and analytical geometry It will be fun 5

6. Setup All patterns are drawn on a square grid. All area is measured by the size of the unit square. 6

7. Calculate simple areas 7 2 x 4 = 8 4×2÷2=4 2×(4 + 2)÷2 = 6

8. Area Formulas Square:a 2 Rectangle:a × b Parallelogram:b × h Triangle:b × h ÷ 2 Trapezoid:(b 1 + b 2 ) × h ÷ 2 8

9. Trapezoid Formula Covers All Trapezoid:(b 1 + b 2 ) × h ÷ 2 Square: a 2 (b 1 = b 2 = h = a) Rectangle: a × b(b 1 =b 2 =a, h=b) Parallelogram: b × h(b 1 = b 2 = b) Triangle: b × h ÷ 2(b 2 = 0, b 1 = b) 9 b1b1 b2b2 h

10. What is this area? 4 x 5 – 3 – 5 – 4 = 8 10 (1) (2) (3) (4)

11. Calculate areas Hat: 3 + 4 + 2 = 9 Goose: 1 + 2 + 4 + 1 = 8 11

12. Cowboy Area = ½ + ½ + 1 + 3 + ½ + ½ + 1 = 7 12

13. Area of a square Area = 5 x 5 – 4 x (2 x 3 / 2) = 25 – 12 = 13 13

14. Length of a side The square’s area is 13, what is the length of its side? The length times itself is 13 It must be more than 3 and less than 4 The value is 3.60555127546…. It is called the square root of 13 with this notation: 13 14

15. There is a relationship How are the three areas of squares related? C = A + B 15 A B C

16. Does the same relationship hold? A = 4, B = 4, is C 8? 16 A B C

17. Pythagorean Theorem The area of the slanted square is always the sum of the area of two straight squares In a right angled triangle the square of the hypotenuse (the longest side of a right triangle) is equal to the sum of the squares of the other two sides. 17

18. Can you explain this? 18

19. Let’s solve the puzzle What is the area of the big triangle we put together? 13 x 5 / 2 = 32.5 19

20. Solving the puzzle What s the area of each colored tiles? Red: 2 x 5 / 2 = 5 Green: 7 Yellow: 8 Blue: 8 x 3 / 2 = 12 Totally: 5 + 7 + 8 + 12 = 32 20

21. Solving the puzzle The top figure covers 32 The lower figure should cover 33 because of the blank Bottom line: both figures are not triangles. The “hypotenuses” are not straight lines. 21

22. Can you explain this? 22

23. Another rectangle This rectangle’s perimeter is 20 and the area is 24. Another rectangle’s area is 20 and perimeter is 24. What is the length and width? 23 4 6

24. Thoughts on the right track You notice that the area is getting smaller but the perimeter is bigger The rectangle has to be “skinnier” to have this The length plus width is 12 So the length is 10 and width is 2 24

25. Magic Here is an interesting number trick done by David Copperfield http://www.youtube.com/watch?v=nZTR7kyz3g8 25

26. If we still have time A 4 x 4 grid is divided into 5 pieces. Fill in the numbers 1, 2, 3, 4 to each of the squares such that the 4 numbers in each column or row are all different and the sum of all digits in each piece are the same. 26

27. Solution Since each column has non repeating numbers each column or row sum to 10 All five pieces have the same sum of digits, they must be all 8 Here is one of the solutions 27 3412 4132 1423 3214