Std 6 Mathematics · Chapter 10

10. Magical Mathematics – 1

Solutions and explanations for all the activities of the chapter. Maharashtra State Board (Balbharati), Grade Six Mathematics.

About the chapter

This chapter shows that mathematics can be fun. It has a number trick with smart cards. We find the angles of a star. We make shapes with pentominoes. We also complete addition-subtraction grids with fractions.

Key points

The first numbers on the smart cards are 1, 2, 4, 8 and 16. Their sum gives the number thought of.
The sum of the angles of a triangle is 180°.
The sum of the angles of a pentagon is 540°. Each angle of a regular pentagon is 108°.
There are 12 different pentominoes. Together they have 60 squares.
In the addition-subtraction grid, the last cell is the same both ways.

Formulas and terms

Pentomino – a shape made by joining 5 equal squares edge to edge.
Regular polygon – a polygon whose all sides and all angles are equal.
Sum of angles of a polygon with n sides = (n − 2) × 180°
Each angle of a regular polygon = (n − 2) × 180° ÷ n

Funny Numbers – 1) Smart Cards

✦How can you find the number your friend is thinking of, just from the cards it appears on?

The first numbers on the five cards are 1, 2, 4, 8 and 16.

Step 1: Ask your friend on which cards his number appears.

Step 2: Add the first numbers of only those cards.

Step 3: The sum is the number your friend is thinking of.

Examples:

23 appears on the cards starting with 16, 4, 2 and 1. 16 + 4 + 2 + 1 = 23
30 appears on the cards starting with 16, 8, 4 and 2. 16 + 8 + 4 + 2 = 30
7 appears on the cards starting with 4, 2 and 1. 4 + 2 + 1 = 7

Every number from 1 to 31 can be made by adding some of 1, 2, 4, 8 and 16. Each number is made in only one way. So the trick always works.

2) Star

✦One angle of the five-pointed star is 72°. Find and state the measures of all the angles in the star.

Every tip of the star measures 36°, each angle of the pentagon inside is 108°, and each base angle of a point is 72°.

Step 1: A regular pentagon is formed inside the star. Each angle of it is 108°.

Step 2: The 72° angle and the 108° angle make a straight angle. 180° − 108° = 72°. So both base angles of each point are 72°.

Step 3: Each point is a triangle. Its tip angle = 180° − (72° + 72°) = 36°.

So, each tip angle of the star is 36°, each base angle of a point is 72° and each angle of the inner pentagon is 108°.

✦Answer the three questions about the star and the polygon inside it.

(1) What is the sum of the measures of all the angles of a pentagon?

A pentagon can be divided into 3 triangles.
Sum of its angles = 3 × 180° = 540°
Each angle of a regular pentagon = 540° ÷ 5 = 108°

(2) If the star has six points, what difference will there be in the measures of the angles?

The polygon inside is a regular hexagon.
Each angle of the hexagon = 4 × 180° ÷ 6 = 120°
Base angle of each point = 180° − 120° = 60°
Tip angle of each point = 180° − (60° + 60°) = 60°

So, in a six-pointed star, each point is an equilateral triangle. The tip angle becomes 60° instead of 36°.

(3) If the star has more than six points, what is the measure of each angle of the polygon inside, and what is the formula for the sum of its angles?

Sum of the angles of a polygon with n sides = (n − 2) × 180°
Each angle of the regular polygon = (n − 2) × 180° ÷ n

7 points – sum = 900°, each angle is about 128.6°
8 points – sum = 1080°, each angle = 135°

3) Pentominoes

✦What are pentominoes, and how many different ones are there?

A pentomino is a shape made by joining 5 equal squares edge to edge. There are 12 different pentominoes. They are named after the English letters they look like.

A shape that is only turned or flipped is the same pentomino.

✦Complete the 5 × 12 rectangle using the pentominoes.

A 5 × 12 rectangle has 60 squares. The 12 pentominoes also have 12 × 5 = 60 squares. So each pentomino is used once. One way to fill it is:

One way of filling the 5 × 12 rectangle — every pentomino used exactly once, with no gaps and no overlaps.

There are many other correct ways. Use all 12 shapes once, with no gaps and no overlaps.

✦What similarity do you see in the different rectangles?

The rectangles are 6 × 10, 5 × 12, 4 × 15 and 3 × 20.

All of them have the same area of 60 squares. Each one is made with all 12 pentominoes. Only the length and breadth change.

4) Addition-Subtraction Grid

✦Complete the two grids. Each row is added across and each column is subtracted downwards.

First grid:

	Column 1	Column 2	Addition →
Row 1	5/3	4/3	9/3 = 3
Row 2	2/3	1/3	3/3 = 1
Subtraction ↓	3/3 = 1	3/3 = 1	2

Row 1 : 5/3 + 4/3 = 9/3 = 3
Row 2 : 2/3 + 1/3 = 3/3 = 1
Column 1 : 5/3 − 2/3 = 3/3 = 1
Column 2 : 4/3 − 1/3 = 3/3 = 1
Last cell : 1 + 1 = 2 and 3 − 1 = 2

Second grid:

	Column 1	Column 2	Addition →
Row 1	1/2	1/3	5/6
Row 2	1/3	1/4	7/12
Subtraction ↓	1/6	1/12	1/4

Row 1 : 1/2 + 1/3 = 3/6 + 2/6 = 5/6
Row 2 : 1/3 + 1/4 = 4/12 + 3/12 = 7/12
Column 1 : 1/2 − 1/3 = 3/6 − 2/6 = 1/6
Column 2 : 1/3 − 1/4 = 4/12 − 3/12 = 1/12
Last cell : 1/6 + 1/12 = 3/12 = 1/4 and 5/6 − 7/12 = 3/12 = 1/4

✦For which other numbers can such grids be made?

Such grids can be made with any numbers, whole numbers or fractions. Each number in the first row should be greater than the number below it.

	Column 1	Column 2	Addition →
Row 1	9	5	14
Row 2	4	2	6
Subtraction ↓	5	3	8

Check : 5 + 3 = 8 and 14 − 6 = 8

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