What is the maximum number of independent meaningful words that "HEARTLESS"…

Concept. Splitting a word this way is a segmentation problem, not an anagram problem: cuts are placed between letters, so the order never changes and no letter is reused. Every piece produced by the cuts must itself be a standalone meaningful English word, and no letter may be left over. The quantity asked for is the largest number of pieces for which this is possible.

Application. The string is H E A R T L E S S - nine letters. Build the split from the left, at each stage asking which prefixes of the remaining letters are words.

1. Prefixes of HEARTLESS that are themselves words: HE, HEAR, HEART and HEARTLESS.

2. HEARTLESS taken whole is one piece.

3. HEART leaves LESS, a word, so HEART + LESS is two pieces; LESS splits no further, since neither L + ESS nor LE + SS gives two words.

4. HEAR leaves TLESS, and none of T, TL, TLE, TLES or TLESS is a word, so this branch stops.

5. HE leaves ARTLESS, whose word prefixes are A, ART and ARTLESS.

6. HE + ARTLESS is two pieces; HE + ART + LESS is three pieces; HE + A leaves RTLESS, and none of R, RT, RTL, RTLE, RTLES or RTLESS is a word.

Cross-check. Every branch has now been followed to its end, so the largest split is three pieces: HE | ART | LESS, using all nine letters once and in their original order.

Contrast. EAR is a meaningful word, but the arrangement H + EAR + T + LESS leaves H and T standing alone, and single letters other than A and I are not English words - so it is not a valid split at all.

Maximum number of independent meaningful words: 3.