(iii) Every part A, B, C, D, E, F, G, H, I, J, K, L, M, N, O and P are 16 triangles.
Combining two at a time. A & B, B & C, C & D, D & A, J & K, L & M, N & O, P & I are 8 triangles.
Taking the combination of 4, we have I, J, E & A; K, L, F and B, M, N, G and C, P, O, H and D are 4 triangles.
Taking 8 parts together we have QRS, RST, STQ, and TRQ are 4 triangles.
\(\therefore \) Total triangles = 16 + 8 + 4 + 4 = 32 triangles.

The smaller portions A, B, C, D, E, F, G and H are 8 triangles.
Combining 2 at a time B & C, D & F, F & G, A & H are 4 triangles.
Taking 4 at a time Combination of B, C, D & E;
D, E, F and G;
F, G, H and A
H, A, Band Care 4 triangles.
Total triangles = 8 + 4 + 4 = 16.

A, B, C, D, & E are five triangles.
No combination of these triangles taken 2 at a time, form any triangle.
Only the combinations of A, F & C; A, F & D; B, F & E; B, F & D and C, F & E form triangles when taken 3 triangles at a time.
Hence, the total number of triangles = 5 + 5 = 10.

A, B, C, D are four triangles.
Combining A & D, B & D and D & C do not form any triangles.
Combining A, B & D; A, C & D and B, D & C also do not form any triangles.
Combining A, B, C & D = 1 triangle
\(\therefore\)Total number of triangles = 4 +1 = 5

We have given two shorts which are black and blue 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 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 shirt are of different colours.
\(\therefore\) We give up Blue short and Blue shirt combination. So we have 5 different combinations.

The single portions  A, B, C, D, E, F, G and H are 8 triangles. 
Taking the combination of 2 E & F, F & G, & H and E & H are 4 triangles.
There are 8 + 4 = 12 triangles.

6 possibilities,
i) R - B - R - B  
ii) R - R - B - B 
iii) B - R - R - B 
iv) B - R - B - R  
v) B - B - R - R  
vi) R - B - B - R
(R- RED, B- BLUE)

(i) The next pattern is 
(ii)

Pattern	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

\(\therefore \) Number of dots in the 25^th pattern = 350.

The given star can be viewed as two magical triangular as.

Now the required arrangement is

Some other arrangements are

Here there are even number of terms.
Also we know that 1 + 6 = 7, 2 + 5 = 7, 3 + 4 = 7;
so placing 7 at the centre, and the pairs (1, 6) (2, 5) and (3, 4) at the opposite ends we get, the 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.

 

 

 

 

 

(i) Yes
(ii) 5
(iii) 
     
Sums are 17, 19, 20, 21 and 23.

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

Step 2: The side having smallest numbers 1 & 2 are to be filled with the greatest number 6, the second smallest 1 & 3 side to be filled with the second largest 5 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 T 3 + 6 = 11.

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