Given a 4x4 matrix with rows I to IV and columns I to IV, letters are placed such that meaningful 4-letter words are formed in each column (top to bottom) and each row (left to right). The coding rule for letters is: A-1, B-2, C-3, D-4, E-5, F-1, G-2, H-3, I-4, ..., Y-5, Z-1. The cell value is the sum of the place values of its row and column. No letter repeats more than twice in the entire matrix. If a letter appears once in a row or column, it must not appear again in that row or column.

Known letters in columns:
- Column I: positions I, II, IV are V, A, T respectively
- Column II: positions III, IV are H, O respectively
- Column III: position II is R
- Column IV: positions I, III, IV are N, T, S respectively

Which alphabet should be placed at position (III, III) to form meaningful words in row III and column III?

Question

Given a 4x4 matrix with rows I to IV and columns I to IV, letters are placed such that meaningful 4-letter words are formed in each column (top to bottom) and each row (left to right). The coding rule for letters is: A-1, B-2, C-3, D-4, E-5, F-1, G-2, H-3, I-4, ..., Y-5, Z-1. The cell value is the sum of the place values of its row and column. No letter repeats more than twice in the entire matrix. If a letter appears once in a row or column, it must not appear again in that row or column.

Known letters in columns:
- Column I: positions I, II, IV are V, A, T respectively
- Column II: positions III, IV are H, O respectively
- Column III: position II is R
- Column IV: positions I, III, IV are N, T, S respectively

Which alphabet should be placed at position (III, III) to form meaningful words in row III and column III?

Accepted Answer

**Solution:**

1. **Given:**
   - Matrix 4x4 with rows I-IV and columns I-IV.
   - Letters coded as A-1, B-2, C-3, D-4, E-5, F-1, G-2, H-3, I-4, ..., Y-5, Z-1.
   - Cell value = row value + column value (both from 1 to 4).
   - No letter repeats more than twice in the entire matrix.
   - If a letter appears once in a row or column, it cannot appear again in that row or column.

2. **Known letters in columns:**
   - Column I: I - V, II - A, IV - T
   - Column II: III - H, IV - O
   - Column III: II - R
   - Column IV: I - N, III - T, IV - S

3. **Goal:** Find letter at (III, III) that makes meaningful words in row III and column III.

4. **Step 1: Determine row and column values:**
   - Row I = 1, II = 2, III = 3, IV = 4
   - Column I = 1, II = 2, III = 3, IV = 4

5. **Step 2: Calculate cell values for (III, III):**
   - Cell value = row + column = 3 + 3 = 6

6. **Step 3: Letter coding and possible letters for cell value 6:**
   - Letters are coded with values 1 to 5 repeatedly (A=1, B=2, ..., E=5, then F=1, etc.)
   - Since cell value is 6, possible letter codes are 6 mod 5 = 1 (since 6 - 5 = 1)
   - So letter code = 1
   - Letters with code 1 are: A, F, K, P, U, Z (since A=1, F=1, K=1, P=1, U=1, Z=1)

7. **Step 4: Check for meaningful words in row III and column III with these letters at (III, III):**

- Column III letters known:
     - II: R
     - III: ? (to find)
     - Others unknown

- Row III letters known:
     - II: ?
     - III: ? (to find)
     - Others unknown

- From the question, column III has R at II and needs a letter at III.

8. **Step 5: Check options given:**
   - (a) U
   - (b) I
   - (c) O
   - (d) R

- From step 6, only letters with code 1 are possible, which includes U but not I, O, or R.

9. **Step 6: Verify no repetition violation:**
   - U appears only once in row III and column III if placed at (III, III).
   - No conflict with other known letters.

10. **Step 7: Conclusion:**
    - The letter at (III, III) must be 'U' to satisfy the coding and form meaningful words.

**Answer:** U

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