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

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

1^st number = 1
2^nd number = 1 + 2 = 3
3^rd number = 3 + 3 = 6
4^th number = 6 + 4 = 10
5^th number = 10 + 5 = 15
6^th number will be 15 + 6 = 21
7^th number will be 21 + 7 = 28
8^th number will be 28 + 8 = 36
9^th number will be 36 + 9 = 45
10^th number will be 45 + 10 = 55
Number of dots in the 10^th figure = 55.

Taking smaller triangles we have AFB, BFG, BGC, CGH, HCD, DHI, DIE, EIJ, EJA and AJF \(\Rightarrow \)10 triangles.
Taking the combination of 2.
CGD, CID, DJE, DHE, EIA, EFA, AGB, AJB, BHC and BFC are 10 triangles.
Combining 3 at a time we have CED, DAE, EBA, ACB, BDC, CEF, DBJ, GDA, EBH, ACI => 10 triangles.
Other triangles are BCE, CAD, DBE, ADB and ACE are 5 triangles.
\(\therefore \) Total 10 + 10 + 10 + 5 = 35 triangles.

(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.

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.

There are 12 straight lines used which is minimum.

There are 7 circles.

(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

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.