In the word "HORIZON", how many distinct pairs of letters are there such that the number of letters between them in the word is the same as the number of letters between them in the English alphabet?

Introduction / Context:
This type of alphabet test question asks you to compare distances between letters in a word with distances between the same letters in the English alphabet. You must count pairs where the number of letters between them in the word matches the number of letters between them in the standard alphabet sequence from A to Z.

Given Data / Assumptions:
• The given word is "HORIZON".
• We consider ordered pairs of letters taken from positions in the word.
• For each pair, we count how many letters lie between them in the word and how many letters lie between their positions in the alphabet.
• A pair is counted if these two counts are equal.

Concept / Approach:
The method is to assign alphabetical positions to each letter (A = 1, B = 2, ..., Z = 26) and also note their positions in the word (1 to 7 for "HORIZON"). For any pair of indices i and j with i < j, the number of letters between them in the word is (j - i - 1). The number of letters between them in the alphabet is the absolute difference of their alphabet positions minus one. We count pairs where these two values match.

Step-by-Step Solution:
Step 1: Write the word and label positions: 1 H, 2 O, 3 R, 4 I, 5 Z, 6 O, 7 N.Step 2: Record alphabet positions: H = 8, O = 15, R = 18, I = 9, Z = 26, N = 14.Step 3: Check pairs systematically.Example pair 1: (H, N) at positions (1,7): letters between them in the word = 7 - 1 - 1 = 5. Alphabet positions are 8 and 14, so letters between them in the alphabet = |14 - 8| - 1 = 5. This pair matches.Example pair 2: (R, O) at positions (3,6): letters between them in the word = 6 - 3 - 1 = 2. Alphabet positions are 18 and 15, so letters between them in the alphabet = |18 - 15| - 1 = 2. This pair also matches.Example pair 3: (R, N) at positions (3,7): letters between them in the word = 7 - 3 - 1 = 3. Alphabet positions 18 and 14 give |18 - 14| - 1 = 3, another match.Example pair 4: (O, N) at positions (6,7): letters between them in the word = 7 - 6 - 1 = 0. Alphabet positions 15 and 14 give |15 - 14| - 1 = 0, yet another match.Step 4: Therefore, we already have four distinct pairs that satisfy the condition.

Verification / Alternative check:
You can continue checking other combinations to ensure that there are no additional valid pairs, but even from these four examples it is clear that the number of qualifying pairs is more than three. Since the answer options group all counts above three into a single choice "More than three", we do not need the exact count beyond confirming that there are at least four such pairs.

Why Other Options Are Wrong:
• One: This would mean only a single pair satisfies the condition, which is contradicted by the four pairs identified above.
• Two: The same reasoning shows that two is too small; we have already found more pairs.
• Three: Again, this is too low, because four or more examples can be verified.
• None: This is clearly incorrect, because even a quick check of (H, N) or (R, O) demonstrates that at least one pair satisfies the distance equality.

Common Pitfalls:
A common mistake is to forget to subtract one when converting the difference in positions into the number of letters between the pair in either the word or the alphabet. Another is to stop after finding one or two valid pairs and assume that the count is complete. Taking a systematic approach by checking all pairs in an organised way avoids these errors.

Final Answer:
In the word "HORIZON", the number of pairs of letters that satisfy the equal distance condition is more than three.