• The line joining P and R is parallel to the line joining Q and S.
• P is placed on the side opposite to the corner T.
• S and U cannot be placed on the same side.
Which one of the following statements is correct based on the above information?
Let the triangle corners be A, B, C and the mid-points of the opposite sides be D, E, F respectively. We are assigning 6 distinct letters (P, Q, R, S, T, U) to these 6 positions.
Analyzing the Conditions
- Condition 1: The line joining P and R is parallel to the line joining Q and S (PR || QS).
- Condition 2: P is placed on the side opposite to the corner T.
- Condition 3: S and U cannot be placed on the same side.
Deduction from Condition 1
The condition PR || QS strongly suggests a relationship based on the Midpoint Theorem. The Midpoint Theorem states that the line segment connecting the midpoints of two sides of a triangle is parallel to the third side. For instance, DE || AB, EF || BC, FD || AC.
If PR || QS, a likely scenario is that P and R represent two corners (e.g., A and B) and Q and S represent the midpoints of the other two sides (e.g., D and E, the midpoints of BC and AC respectively). In this case, the line segment DE is parallel to the side AB (which connects the two corners P and R).
Therefore, we can hypothesize:
- {P, R} = {Corners A, B}
- {Q, S} = {Midpoints D, E} (midpoints of sides opposite A and B)
Deduction from Condition 2
P is on the side opposite to corner T.
Based on the hypothesis from Condition 1, P is a corner (A or B). The side opposite a corner is the side connecting the other two corners. If P=A and R=B, the third corner is C. The side opposite C is AB.
Since P (A or B) is one of the endpoints of the side AB, P lies on the side opposite corner C. This implies that T must be the third corner (C).
So, T = C.
Determining Positions of Remaining Letters
Letters used so far: P, R (corners A, B), T (corner C), Q, S (midpoints D, E).
The only remaining letter is U. The only remaining position is the midpoint of the side opposite T (corner C), which is the midpoint of AB. Let this be F.
Therefore, U = F (midpoint of AB).
Checking Condition 3
S and U cannot be placed on the same side.
We have U = F (midpoint of AB). S is either D (midpoint of BC) or E (midpoint of AC).
- If S=D, U=F. S is on side BC, U is on side AB. They are not on the same side segment.
- If S=E, U=F. S is on side AC, U is on side AB. They are not on the same side segment.
Condition 3 holds true in this configuration.
Summary of a Valid Configuration
A valid assignment satisfying all conditions is:
- Corners: {P, R, T} = {A, B, C}
- Midpoints: {Q, S, U} = {D, E, F} (where {Q, S} are midpoints of sides adjacent to T, and U is the midpoint of the side opposite T)
In this configuration, S is assigned to a midpoint (either D or E).
Evaluating the Options
- P cannot be placed at a corner: This is false, as our valid configuration places P at a corner.
- S cannot be placed at a corner: This is true, as our valid configuration places S at a midpoint.
- U cannot be placed at a mid-point: This is false, as our valid configuration places U at a midpoint (F).
- R cannot be placed at a corner: This is false, as our valid configuration places R at a corner.
Based on the consistent deduction, S must be a midpoint.
Important Questions from Puzzles
In the 4 x 4 array shown below, each cell of the first three rows has either a cross (X) or a number.
1 X 4 3 X 5 5 4 3 X 6 X
The number in a cell represents the count of the immediate neighboring cells (left, right, top, bottom, diagonals) NOT having a cross (X). Given that the last row has no crosses (X), the sum of the four numbers to be filled in the last row isStudents applying for hostel rooms are allotted rooms in order of seniority. Students already staying in a room will move if they get a room in their preferred list. Preferences of lower ranked applicants are ignored during allocation.
Given the data below, which room will Ajit stay in?Names Student seniority Current room Room preference list Amar 1 P R, S, Q Akbar 2 None R, S Anthony 3 Q P Ajit 4 S Q, P, R The diagram below shows a river system consisting of 7 segments, marked P, Q, R, S, T, U, and V. It splits the land into 5 zones, marked Z1, Z2, Z3, Z4, and Z5. We need to connect these zones using the least number of bridges. Out of the following options, which one is correct?
Note: The figure shown is representative.
In the square grid shown on the left, a person standing at P2 position is required to move to P5 position.
The only movement allowed for a step involves, “two moves along one direction followed by one move in a perpendicular direction”. The permissible directions for movement are shown as dotted arrows in the right.
For example, a person at a given position Y can move only to the positions marked X on the right.
Without occupying any of the shaded squares at the end of each step, the minimum number of steps required to go from P2 to P5 is
- Three individuals are named P, Q, and R. Together they have a total of fifteen children of which nine are boys. P has three girls and Q has same number of boys. Q has one more child than P, who has four children. R has four more boys than the number of girls. The number of girls of R is equal to the number of boys of P. How many boys do R and P have?
