On a special clock, the usual numbers 1 to 12 are replaced by letters in order: the number 1 is replaced by M, the number 2 by N, the number 3 by O, and so on around the dial. When the time is 21:00 hours (9 p m in the evening), on which letter will the hour hand of this coded clock be pointing?

Introduction / Context:
This question blends time reading on a clock with a simple substitution of letters for numbers. Instead of showing numbers from 1 to 12, the clock uses letters starting from M at the position of 1 and continuing in alphabetical order. We are asked to find the letter corresponding to the hour hand position at 21:00 hours, which is the same as 9 p m in the usual 12 hour format. This tests basic understanding of time conversion and ordered mapping of numbers to letters.

Given Data / Assumptions:

• The clock face has letters instead of the numbers 1 to 12.
• Number 1 is replaced by M.
• Number 2 is replaced by N.
• Number 3 is replaced by O and the pattern continues alphabetically around the dial.
• We are told to consider the time 21:00 hours, which corresponds to 9 p m.
• The hour hand at 21:00 points to the same position as at 9 o clock in the 12 hour system.

Concept / Approach:
The idea is to map the 12 hour positions to letters. Since 1 is M and each next number gets the next alphabet letter, we can list the mapping from 1 to 12. After that, we convert the 24 hour time 21:00 into its 12 hour equivalent. Finally, we match the hour position with the correct letter. The arithmetic involved is very simple and the main challenge is to keep the mapping and the time conversion straight in our mind.

Step-by-Step Solution:
Step 1: List the letters corresponding to each clock number. Starting at 1 we have: 1 → M, 2 → N, 3 → O, 4 → P, 5 → Q, 6 → R, 7 → S, 8 → T, 9 → U, 10 → V, 11 → W, 12 → X. Step 2: Convert 21:00 hours into a 12 hour format. Subtract 12 from 21 to get 9, so 21:00 is the same as 9 p m. Step 3: At 9 p m, the hour hand points to the 9 on a standard clock. Step 4: From the mapping above, the number 9 is replaced by the letter U. Step 5: Therefore, at 21:00 the hour hand will be pointing at the letter U on this coded clock.

Verification / Alternative check:
As a simple check, we can look at a few positions to ensure the mapping is consistent. The first few letters M, N, O clearly follow 1, 2, 3. By continuing the alphabetical sequence, we see that 8 is T and 9 is U, which fits smoothly in the pattern. There is no break or reset in the letter sequence, so our mapping for 9 as U is secure. The conversion 21:00 to 9 o clock is a standard 24 hour to 12 hour conversion and is also straightforward.

Why Other Options Are Wrong:
Option T corresponds to the position of 8, which would be correct for 20:00 hours but not for 21:00. Option S corresponds to 7, and V corresponds to 10. Option R would map to 6. None of these positions represent 9 o clock on the dial, so they are incorrect for 21:00 hours. Only U matches the position of the hour hand at that time when we use the given letter mapping.

Common Pitfalls:
A common mistake is to misconvert 21:00 hours and think of it as 8 p m or 10 p m. Another frequent error is to offset the letters incorrectly, for example starting from L instead of M at 1 or skipping a letter in the sequence. Some learners also confuse the letter at 9 with the one at 8 because both are near the top of the dial in their mental picture. Writing down the full mapping from 1 to 12 with the letters beside them avoids these problems.

Final Answer:
At 21:00 hours the hour hand points to the 9 on a normal clock, which is coded as the letter U on this special clock.