Let \(C\) represent completing the puzzle and \(C'\) represent not completing it.

The probability that Kenny completes the puzzle on both days is \(P(C, C) = 0.8 \times 0.9 = 0.72\).

The probability that Kenny completes the puzzle on Monday but not on Tuesday is \(P(C, C') = 0.8 \times 0.1 = 0.08\).

The probability that Kenny does not complete the puzzle on Monday but completes it on Tuesday is \(P(C', C) = 0.2 \times 0.6 = 0.12\).

The probability that Kenny completes the puzzle on at least one of the two days is:

\(P(C, C) + P(C, C') + P(C', C) = 0.72 + 0.08 + 0.12 = 0.92\)

Alternatively, the probability that Kenny does not complete the puzzle on both days is \(P(C', C') = 0.2 \times 0.4 = 0.08\).

Thus, the probability that he completes the puzzle on at least one day is:

\(1 - P(C', C') = 1 - 0.08 = 0.92\)