Seven men A, B, C, D, E, F and G are standing in a queue, one behind the other in that order. That is, A is first in queue while G brings up the rear. Each of them is wearing a hat of a different colour of the rainbow - Violet, Indigo, Blue, Green, Yellow, Orange, and Red, but not necessarily in that order. It is also known that:
• D can see in front of him green and blue hats, but not the violet hat.
• E can see violet and yellow hats, but not the red hat.
• G can see hats of all the colours except Orange.
• E is wearing the Indigo coloured hat.
What colour hat is worn by F?

Question

Prepp Team · Accepted Answer

Queue Setup and Colors

Seven people (A, B, C, D, E, F, G) are in a queue, ordered from front to rear (A is first, G is last).

They wear hats of distinct rainbow colours:

Violet ($V$)
Indigo ($I$)
Blue ($B$)
Green ($G$)
Yellow ($Y$)
Orange ($O$)
Red ($R$)

Hat Color Deductions

Key clues are used for step-by-step deduction to find the hat colours.

G Hat Color

Clue: G sees all hats except Orange ($O$). G sees people A-F.

Reasoning: Since G sees everyone in front (A to F) and observes all colours but Orange, G must be wearing the Orange ($O$) hat.

Result: Hat($G$) = $O$.

E Hat Color

Clue: E is wearing the Indigo ($I$) hat.

Result: Hat($E$) = $I$.

F Hat Color

Clue: E sees Violet ($V$) and Yellow ($Y$) hats, but not the Red ($R$) hat. E sees people A, B, C, D.

Reasoning: E not seeing Red ($R$) means $R$ is not worn by A, B, C, or D. Therefore, $R$ must be worn by E, F, or G.

Knowing Hat($E$) = $I$ and Hat($G$) = $O$, the Red ($R$) hat must belong to F.

Result: Hat($F$) = $R$.

D Hat Color

Clue: D sees Green ($G$) and Blue ($B$) hats, but not the Violet ($V$) hat. D sees people A, B, C.

Reasoning: D not seeing Violet ($V$) means $V$ is not worn by A, B, or C. Therefore, $V$ must be worn by D, E, F, or G.

Since Hat($E$) = $I$, Hat($F$) = $R$, and Hat($G$) = $O$, the Violet ($V$) hat must belong to D.

Result: Hat($D$) = $V$.

Remaining Colors Assignment

The hats assigned so far are: $D=V$, $E=I$, $F=R$, $G=O$.

The remaining people are A, B, C.

The remaining colours are Blue ($B$), Green ($G$), and Yellow ($Y$).

These colours ($B, G, Y$) must be worn by A, B, and C.

Clue Verification

The deduced assignments are consistent with all the given clues:

D (wears $V$) sees the hats of A, B, C. The clue states D sees $G$ and $B$ hats, and does not see $V$. This is consistent as $V$ is worn by D himself.
E (wears $I$) sees the hats of A, B, C, D. The clue states E sees $V$ (worn by D) and $Y$ (worn by A, B, or C), and does not see $R$ (worn by F). This is consistent.
G (wears $O$) sees the hats of A, B, C, D, E, F. These hats are $\{B, G, Y, V, I, R\}$, which are all colours except Orange ($O$). This is consistent.

The deduction process determines the hat colors for D, E, F, and G based on the provided information.