(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.

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.

(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 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.

 

 

 

 

 

The number of dots given are 1, 4, 9, 11, 16, 25, ...
The no. of dots are in the pattern 1 \(\times\) 1, 2 \(\times\) 2, 3 \(\times\) 3, 4 \(\times\) 4, 5 \(\times\) 5 .... 10 \(\times\) 10
\(\therefore \) The number of dots in the 10^th figure = 100.

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.