Completing a letter-number-symbol matrix
The following 3×3 matrix is built on consistent row and column rules. Each cell contains a letter, a number, and one of three symbols: ▲, ●, ■.
| C1 | C2 | C3 | |
|---|---|---|---|
| R1 | H4▲ | K8● | O13■ |
| R2 | J6● | M10■ | Q15▲ |
| R3 | L8■ | O12▲ | ? |
The rules are: letters advance by a fixed step across each row; numbers advance by a fixed step across each row; each row and each column contains each symbol exactly once. The same letter-step and number-step patterns apply in every row. No cell repeats a letter-number pair. The matrix is complete except for R3C3. A colleague proposes S17▲. Another proposes S19●. You must decide which, if either, is consistent, and explain the rule you used.
1. Complete the missing cell. Show the letter, number, and symbol reasoning separately.
Read the worked answer
Letter: In R1, H → K is +3 and K → O is +4. In R2, J → M is +3 and M → Q is +4. So R3 follows L → O (+3), then O → S (+4). Number: In R1, 4 → 8 is +4 and 8 → 13 is +5. In R2, 6 → 10 is +4 and 10 → 15 is +5. So R3 follows 8 → 12 (+4), then 12 → 17 (+5). Symbol: R3 already contains ■ in C1 and ▲ in C2, so C3 must be ● if each row contains each symbol once. Column C3 already contains ■ and ▲, which confirms ●. The missing cell is therefore S17●.
2. Evaluate the colleague’s proposal S17▲. Which rule does it violate?
Read the worked answer
S17▲ has the correct letter S and the correct number 17, but it violates the symbol rule. Row R3 would then contain ■, ▲, ▲: two triangles and no circle. Column C3 would contain ■, ▲, ▲: again a duplicate and no circle. The rule states that each row and each column contains each symbol exactly once, so S17▲ is inconsistent. The other proposal, S19●, has the correct symbol but the wrong number: the number step in R3 is +4 then +5, giving 17, not 19. S19● would require a different and inconsistent number pattern.
3. If the symbol rule were removed, would the answer still be unique? Explain.
Read the worked answer
Without the symbol rule, the letter and number would still be uniquely determined as S17, because the letter steps (+3, +4) and number steps (+4, +5) are consistent across all rows. But the symbol would not be determined by the remaining rules: with no row or column constraint, any of ▲, ●, or ■ could accompany S17. The full answer S17● depends on the symbol Latin-square condition. This is why the three rules must be applied separately: the letter and number rules fix the alphanumeric part, while the symbol rule fixes the final character.