In each of the following questions, the second word is formed by rearranging…

Concept

A letter-rearrangement analogy is governed by a positional rule rather than by the particular letters involved: the first pair of words fixes a permutation of place numbers, stating which place each letter travels to. Read that permutation by numbering the letters of the source word and then recording the place each numbered letter occupies in its transformed partner. The same permutation is applied place by place to the third word, so every admissible answer must be an anagram of that third word.

Applying the rule here

1. Number the letters of NUMBER: N(1), U(2), M(3), B(4), E(5), R(6).

2. Write UNBMRE carrying those same tags: U(2), N(1), B(4), M(3), R(6), E(5).

3. The place map is therefore 1 with 2, 3 with 4, and 5 with 6: each adjacent pair of places exchanges its two letters, and no letter leaves its own pair.

4. Number the letters of GHOSTS: G(1), H(2), O(3), S(4), T(5), S(6).

5. Exchange the letters inside each adjacent pair: (G, H) becomes H, G; (O, S) becomes S, O; (T, S) becomes S, T.

6. Reading the six places in order gives H, G, S, O, S, T, that is HGSOST.

Cross-check

A pairwise exchange is its own inverse, so carrying it out a second time returns the starting word. Exchanging the adjacent pairs of HGSOST gives back G, H, O, S, T, S, which is GHOSTS, and carrying out that same single exchange on UNBMRE likewise returns NUMBER. HGSOST is also an anagram of GHOSTS, using one G, one H, one O, one T and two S, as any admissible answer must.

Common slips

• Carrying out only the opening exchange, which moves G and H but leaves O, S, T and S untouched.

• Rotating the opening three letters instead of exchanging them in pairs, which sends G away to the third place.

• Exchanging the first two pairs and then stopping, so the closing T and S are never exchanged.