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.

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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?

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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?

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Olympiad Math III

Lesson 10Area of patterns on grids

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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

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Setup

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

square.

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Calculate simple areas

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2 x 4 = 8

2 x 4 = 8

4×2÷2=4

2×(4 + 2)÷2 = 6

Area Formulas

• Square: a2

• Rectangle: a × b• Parallelogram: b × h• Triangle: b × h ÷ 2• Trapezoid: (b1 + b2) × h ÷ 2

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Trapezoid Formula Covers All

• Trapezoid: (b1 + b2) × h ÷ 2

• Square: a2 (b1 = b2 = h = a)

• Rectangle: a × b (b1=b2=a, h=b)

• Parallelogram: b × h (b1 = b2 = b)

• Triangle: b × h ÷ 2 (b2 = 0, b1 = b)9

b1

b2

h

What is this area?

4 x 5 – 3 – 5 – 4 = 8

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(1)

(2)

(3)

(4)

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

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Cowboy• Area = ½ + ½ + 1 + 3 + ½ + ½ + 1 = 7

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Area of a square

• Area = 5 x 5 – 4 x (2 x 3 / 2) = 25 – 12 = 13

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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

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There is a relationship

• How are the three areas of squares related?

• C = A + B15

A

B

C

Does the same relationship hold?

• A = 4, B = 4, is C 8?

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A

BC

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.

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Can you explain this?

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Let’s solve the puzzle

• What is the area of the big triangle we put together?

• 13 x 5 / 2 = 32.5

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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

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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.

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Can you explain this?

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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?

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4

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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

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Magic

• Here is an interesting number trick done by David Copperfield

http://www.youtube.com/watch?v=nZTR7kyz3g8

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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.

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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

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3 4 1 2

41 3 2

14 2 3

32 1 4