In a certain coding scheme, 'AC' is coded as 8 and 'PQ' is coded as 33. What…

Concept: in alphabet–number coding every letter is first replaced by its serial position in the English alphabet (A = 1, B = 2, … , Z = 26), and the code of a letter pair is then produced by one fixed arithmetic operation applied to those two position numbers. That operation is never stated in the question; it has to be inferred from the solved pairs and then applied unchanged to the pair being asked about.

Application: convert the solved pairs into position numbers and test operations until one rule fits every clue.

1. Write down the positions of the letters in the given pairs: A = 1, C = 3, P = 16 and Q = 17.

2. Test the plain sum first: 1 + 3 = 4, while the code given for 'AC' is 8, so the sum by itself cannot be the rule.

3. Note the gap inside each pair: 3 − 1 = 2 for 'AC' and 17 − 16 = 1 for 'PQ'.

4. Multiply the sum by the gap: (1 + 3) × (3 − 1) = 4 × 2 = 8 and (16 + 17) × (17 − 16) = 33 × 1 = 33, which reproduces both codes given in the question.

5. Apply that same rule to 'XZ', where X = 24 and Z = 26: (24 + 26) × (26 − 24) = 50 × 2 = 100.

Cross-check: the identity below shows that this rule is simply the difference of the squares of the two alphabet positions, and it reproduces every clue.

(a + b) × (b − a) = b2 − a2

• 32 − 12 = 9 − 1 = 8

• 172 − 162 = 289 − 256 = 33

• 262 − 242 = 676 − 576 = 100

Both codes stated in the question come out right under this reading, so the code for 'XZ' is 100.