Directions (83-87): Study the following information carefully and answer the questions given below-

Six persons are living on different floors of six storey building. Ground floor is numbered as 1st floor and just above the floor is numbered as 2nd floor and so on till the topmost floor is numbered as 6th floor. For Diwali Each of them made rangoli on their floor in different colours. B lives on an even numbered floor but neither made rangoli in Hazel nor purple colour. Three persons live between B and the one who made rangoli in green colour. One person lives between the ones who made rangoli in green and blue colour. E lives below the one who made rangoli in blue colour but not just below. E does not make rangoli in purple and green colour. The number of persons live below A is same as the number of persons live above E. D lives above the one who made rangoli in purple colour but not just above. The one who made rangoli in yellow colour lives above the one who made rangoli in violet colour. C lives above F. E does not make rangoli in violet colour .

Which of the following is true?
I. A does not live below the one who made rangoli in yellow
II. C does not make rangoli in hazel colour
III. E lives on the 2nd floor

The correct answer is Both I and II.

Step-by-Step Explanation and Derivation:

Let us analyze the given information step-by-step to determine the floor arrangement (from 6th floor down to 1st floor) and the corresponding rangoli colors for each person.

Given Details:

1. A 6-storey building (Floors: 6, 5, 4, 3, 2, 1).

2. Six persons: A, B, C, D, E, F.

3. Six rangoli colors: Green, Blue, Yellow, Violet, Hazel, Purple.

Step 1: Determine the positions of B, Green, and Blue

- B lives on an even-numbered floor (Floor 6, 4, or 2).

- Three persons live between B and the one who made rangoli in green color.

- If B is on Floor 6, Green is on Floor 2.

- If B is on Floor 2, Green is on Floor 6.

- (B cannot be on Floor 4 because 3 floors gap would be impossible).

- One person lives between the ones who made rangoli in green and blue color.

- If Green is on Floor 2, Blue can be on Floor 4 (or Floor 0, which is invalid).

- If Green is on Floor 6, Blue is on Floor 4.

- E lives below the one who made rangoli in blue color, but NOT just below.

- E does not make green or purple rangoli.

- B does not make hazel or purple rangoli.

Step 2: Testing Cases for B and E

Case 1: B is on Floor 6

- B = Floor 6 (Colors not Hazel/Purple)

- Green = Floor 2

- Blue = Floor 4 (since 1 person lives between Green and Blue)

- E lives below Blue (Floor 4) but not just below (so E cannot be on Floor 3).

- E also cannot be Green (Floor 2). Thus, E must be on Floor 1.

- The number of persons living below A equals the number of persons living above E.

- Number of persons below E (Floor 1) = 0.

- Therefore, persons above E = 5. So persons below A must be 5, which puts A on Floor 6. But B is already on Floor 6! So Case 1 is invalid.

Case 2: B is on Floor 2

- B = Floor 2 (Colors not Hazel/Purple)

- Green = Floor 6 (3 persons between Floor 2 and Floor 6: Floors 3, 4, 5)

- Blue = Floor 4 (1 person between Floor 6 and Floor 4)

- E lives below Blue (Floor 4) but NOT just below (so E cannot be on Floor 3).

- E is not on Floor 2 (since B is on Floor 2).

- E is not Green (Floor 6).

- Hence, E must be on Floor 1.

Step 3: Determining positions of A, C, D, F

- E is on Floor 1. Number of persons living above E = 5.

- Number of persons living below A = 5 ⇒ A is on Floor 6 (Green rangoli).

- Since C lives above F, and D lives above the one with Purple rangoli (not just above):

- The remaining available floors for C, D, F are Floors 5, 3, and 1 (since Floor 6 = A, Floor 4 = Blue, Floor 2 = B).

- E is on Floor 1, so the remaining persons for Floors 5 and 3 are C, D, and F. But there are only two remaining slots (Floors 5 and 3) for persons C, D, F. Let's re-verify:

- Person on Floor 1 = E.

- Floor 2 = B.

- Floor 4 = (Blue rangoli).

- Floor 6 = A (Green rangoli).

- Floors 3 and 5 are left for C, D, F. Wait, D must be above purple. The remaining persons are C, D, F. D can be on Floor 5, C on Floor 4, F on Floor 3.

- Let's check Floor 4: Blue rangoli. C can live on Floor 4 (Blue)!

- Then D is on Floor 5, and F is on Floor 3.

- Check condition: "C lives above F" ⇒ C (Floor 4) > F (Floor 3). This is satisfied.

Step 4: Determining Colors

- Floor 6: A = Green

- Floor 5: D

- Floor 4: C = Blue

- Floor 3: F

- Floor 2: B

- Floor 1: E

- "D lives above the one who made rangoli in purple colour but not just above":

- D is on Floor 5. Purple can be on Floor 2 or Floor 1 (since it can't be Floor 4 which is just below D).

- E (Floor 1) does not make purple rangoli. B (Floor 2) does not make purple rangoli. So Purple must be on Floor 3 (F = Purple)!

- Wait, Floor 4 is just below Floor 5. Floor 3 is not just below Floor 5. So F (Floor 3) = Purple.

- "The one who made rangoli in yellow colour lives above the one who made rangoli in violet colour":

- E does not make violet rangoli. So E (Floor 1) cannot be Violet.

- If Yellow is on Floor 5 (D), Violet is on Floor 2 (B).

- Then Hazel is on Floor 1 (E). (Since B cannot be Hazel, this fits: B = Violet, E = Hazel).

Final Arrangement Table:

- Floor 6: A — Green

- Floor 5: D — Yellow

- Floor 4: C — Blue

- Floor 3: F — Purple

- Floor 2: B — Violet

- Floor 1: E — Hazel

Evaluating the Statements:

I. A does not live below the one who made rangoli in yellow:

A is on Floor 6 and Yellow is on Floor 5. A lives above Yellow, so A does not live below Yellow. (TRUE)

II. C does not make rangoli in hazel colour:

C makes Blue rangoli, which means C does not make Hazel rangoli. (TRUE)

III. E lives on the 2nd floor:

E lives on the 1st floor, not the 2nd floor. (FALSE)

Hence, statements I and II are true.