5 marks

Question 15 Marks

Find the number of dots in the tenth figure of the following pattern.

Answer

The number of dots given is 1, 3, 6, 10, 15 ...

Pattern	Number of dots
1	1
2	3
3	6
4	10
5	15

$1^{\text {st }}$ number $=1$
$2^{\text {nd }}$ number $=1+2=3$
$3^{\text {rd }}$ number $=3+3=6$
$4^{\text {th }}$ number $=6+4=10$
$5^{\text {th }}$ number $=10+5=15$
$6^{\text {th }}$ number will be $15+6=21$
$7^{\text {th }}$ number will be $21+7=28$
$8^{\text {th }}$ number will be $28+8=36$
$9^{\text {th }}$ number will be $36+9=45$
$10^{\text {th }}$ number will be $45+10=55$
Number of dots in the $10^{\text {th }}$ fiqure $=55$

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Question 25 Marks

Place the number 1 to 12 in the 12 circles so that the sum of the numbers in the six lines of the star is 26. Use each number from 1 to 12 exactly once. Find more possible ways?

Answer

The given star can be viewed as two magical triangular as

Now the required arrangement is

Some other arrangements are

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Question 35 Marks

Arrange the odd numbers from 1 to 17 without repetition to get a sum of 30 on each side of the magic triangle.

Answer

The odd numbers between 1 to 17 are $1,3,5,7,9,11,13,15,17$.
Step 1: Place the smaller numbers 1, 3, 5 on the comers.

Step 2: Arrange another set of smaller numbers 7, 9 and 11 on each side.

Step 3: Arrange the remaining numbers $13,15,17$ to give the total 30 .

Magic sum $=30$.

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Question 45 Marks

Using the numbers from 1 to 9
(i) Can you form a magic triangle?
(ii) How many magic triangles can be formed?
(iii) What are the sums of the sides of the magic triangle?

Answer

(i) Yes, we can form
(ii) 5
(iii)

Sums are 17, 19, 20, 21 and 23.

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Question 55 Marks

In the following magic triangle, arrange the numbers from 1 to 6, so that you get the same sum on all its sides.

Answer

Step 1: Complete the corners with smaller numbers 1, 2 and 3.

Step 2: The side having smallest numbers 1 and 2 are to be filled with the greatest number 6 , the second smallest 1 and 3 side to be filled with the second largest 5 at the middle and so on. at the middle and so on.

The magic sum is $1+6+2=2+4+3=3+5+1=9$.
Some other ways are given below.

The magic sum = 1 + 6 + 3 = 3 + 2 + 5 = 5 + 4 + 1 = 10.

The magic sum 6 + 1 + 4 = 4 + 5 + 2 = 2 + 3 + 6 = 11.

The magic sum 4 + 3 + 5 = 5 + 1 + 6 = 6 + 2 + 4 = 12.

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Question 65 Marks

Suppose, you have two shorts, one is black and the other one is blue; three shirts which are in white, blue and red. You again wish to make different combinations, but you always want to make sure that the shorts and shirt that you wear are of different colours. List and check how many combinations are possible now

Answer

We have given two shorts which are black and blue in colour.
Take it as T black and T blue.
Also, we have 3 shirts, coloured white, blue and red denoted by S white, S blue and S red.
Now fix T black and then T blue the different combinations are

Thus we get a total of 6 combinations as
Black short and White shirt
Black short and Blue shirt
Black short and Red shirt
Blue short and White shirt
Blue short and Blue shirt
Blue short and Red shirt.
But it is given short and the shirt is of different colours.
We give up Blue short and Blue shirt combination.
So we have 5 different combinations.

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Question 75 Marks

(i) Draw the next pattern
(ii) Prepare a table for the number of dots used for each pattern
(iii) Explain the pattern
(iv) Find the number of dots in the $25^{th}$ pattern

Answer

(i) The next pattern is

(ii)

Pattern Number	1	2	3	4	5
Number of dots	2	5	9	14	20

(iii)

Pattern	Number of dots
1	2
2	2 + 3
3	2 + 3 + 4
4	2 + 3 + 4 + 5

First number is 2
$2^{nd}$ number is 2 + 3 = 5
$3^{rd}$ number is 5 + 4 = 9
$|4^{th}$ number is 9 + 5 = 14
$5^{th}$ number is 14 + 6 = 20(iv) Number of dots in the $25^{th}$ pattern is

Pattern	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25
No. of dots	2	5	9	14	20	27	35	44	54	65	77	90	104	119	135	152	170	189	209	230	252	275	299	324	350

The number of dots in the $25^{th}$ pattern = 350.

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