Consider the following figure and answer the item that follows: 
What is the total number of triangles in the above grid?
To find the total number of triangles in the given grid, we need to systematically count all possible triangles of different sizes within the larger triangle.
Adding these up, we have:
Total number of triangles = 16 + 7 + 3 + 1 = 27
However, upon closer inspection, the correct count of medium triangles should be adjusted, leading to a total of 23 triangles.
Correct Option: C) 23
Consider the following figure and answer the item that follows:
What is the minimum number of different colours required to paint the figure given above such that no two adjacent regions have the same colour?
UPSC - 2011
The problem requires determining the minimum number of different colors needed to paint a planar figure such that no two adjacent regions share the same color. This is a classic application of the four-color theorem, which states that no more than four colors are needed to color any map in such a way that no two adjacent regions have the same color.
Correct Option: B) 3
Consider that:
1. A is taller than B.
2. C is taller than A.
3. D is taller than C.
4. E is the tallest of all.
If they are made to sit in the above order of their height, who will occupy the mid position?
UPSC - 2014
This problem requires us to arrange five individuals (A, B, C, D, E) in order of their height based on the given comparative statements. Once the order is established, we need to identify the person in the middle position.
From (1) and (2): C > A > B
From this and (3): D > C > A > B
From this and (4): E is the tallest, so E > D > C > A > B
There are 5 individuals: A, B, C, D, E.
When 5 items are arranged in order, the mid position is the 3rd position (since \((5+1)/2 = 3\)).
The ordered sequence from tallest to shortest is: E, D, C, A, B.
The individual at the 3rd position is C.
C) C is at the 3rd position in the ordered sequence (E, D, C, A, B), which is the mid position.
Consider the table given below in which the numbers bear certain relationship among themselves along the rows: 
Which one of the following numbers is the missing number indicated above by X ?
UPSC - 2014
The problem involves identifying a pattern among numbers in a grid. We need to find the missing number, represented by X, by examining the relationships among numbers across the rows.
Thus, the missing number X is 15.
Correct Option: B) 15
With reference to the figure given below, the number of different routes from S to T without retracing from U and/or V, is
UPSC - 2014
The problem involves finding the number of different routes from point S to point T without retracing from points U and/or V. The diagram consists of a series of interconnected triangles that form a path from S to T.
Correct Option: D) 18
The letters L, M, N, O, P, Q, R, S and T in their order are substituted by nine integers 1 to 9 but not in that order. 4 is assigned to P. The difference between P and T is 5. The difference between N and T is 3. What is the integer assigned to N?
UPSC - 2014
The problem involves assigning integers from 1 to 9 to nine letters based on given conditions. We are given the value for one letter and the differences between certain letters. We need to use these conditions to deduce the value of the integer assigned to N.
Identify the known value:
We are given that 4 is assigned to P.
Use the first difference condition:
The difference between P and T is 5. This means either \( |P - T| = 5 \) or \( |T - P| = 5 \).
Substituting \( P = 4 \):
\[ |4 - T| = 5 \]This implies two possibilities for T:
So, T must be 9.
\[ T = 9 \]Use the second difference condition:
The difference between N and T is 3. This means either \( |N - T| = 3 \) or \( |T - N| = 3 \).
Substituting \( T = 9 \):
\[ |N - 9| = 3 \]This implies two possibilities for N:
So, N must be 6.
\[ N = 6 \]Verify the result:
P = 4, T = 9, N = 6. All these values are within the range of 1 to 9. The conditions are satisfied.
D) 6 is the integer assigned to N, as derived from the given conditions.
