Each vowel in the word POLARISE is changed to the letter immediately following…

Concept — In a letter-shift code every letter is replaced through its position in the English alphabet: a forward shift of one step gives the next letter, while a backward shift of two steps gives the letter two places earlier. When vowels and consonants are governed by different shift rules, classify each letter first and then apply only the rule that belongs to it. Counting repetitions is a separate, final step: tally the coded letters and read off how many distinct letters occur with the required frequency.

Application — Apply the two rules to POLARISE, letter by letter.

1. Write POLARISE letter by letter in order: P, O, L, A, R, I, S, E.

2. Classify each letter: the vowels are O, A, I and E; the consonants are P, L, R and S.

3. Move every vowel one place forward and every consonant two places back, as the table below sets out.

4. Read the coded letters in the original order: N, P, J, B, P, J, Q, F.

5. Tally the coded letters: N once, P twice, J twice, B once, Q once and F once.

6. Exactly two coded letters, P and J, occur twice, so the required count is two.

Cross-check — The two repeated letters occupy 4 of the 8 coded positions and the remaining four letters occupy one position each, so 4 + 4 = 8 matches the length of POLARISE. Each repeat is traceable to two different sources: the coded P arises from the vowel O and from the consonant R, while the coded J arises from the vowel I and from the consonant L. The original P does not survive as P — it becomes N.

Common slips —

• Shifting the consonants two places forward instead of backward gives R, P, N, B, T, J, U, F, in which every letter is different and nothing repeats.

• Shifting the vowels one place backward instead of forward gives N, N, J, Z, P, H, Q, D, in which only N repeats.

Hence exactly two letters, P and J, appear twice in the new group of letters.