From reader Éric Angelini:

Call this S₁:

FIRST, SECOND, THIRD, FOURTH, FIFTH, SIXTH, SEVENTH, EIGHTH, NINTH, TENTH, ELEVENTH, …

Consider it both a string of letters and a list of instructions: We are to underline the indicated letters, in order. That’s pretty straightforward — we’ll underline the first letter, then the second, then the third, and so on, ultimately reproducing S₁.

Suppose we start the list with SIXTH, rather than FIRST. Now our first instruction is to underline the sixth letter, which is the S in SECOND. After that we underline the second letter in the string, as before, and the third, and so on. Only the very first letter, the F in FIRST, has been overlooked, and we can remedy that by putting FIRST in the sixth position in the list. With that swap all is well:

S₆ = SIXTH, SECOND, THIRD, FOURTH, FIFTH, FIRST, SEVENTH, EIGHTH, NINTH, TENTH, ELEVENTH, …

Similarly, if the list starts with EIGHTH we can get everything underlined with just a single swap:

S₈ = EIGHTH, SECOND, THIRD, FOURTH, FIFTH, SIXTH, SEVENTH, FIRST, NINTH, TENTH, ELEVENTH, …

Éric asks, “What about starting S₉ with NINTH? How many swaps do we need to reproduce S₉? This is more tricky!”

See the answer by Hans Havermann at the bottom of this page.

(Thanks, Éric.)