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Annie's scores for the four rounds were 61, 67, 71 and 70.

Bonnie's scores were 62, 65, 74 and 77.

Connie's scores were 63, 68, 72 and 75.

Donnie's scores were 64, 66, 70 and 78.

Here is one solution:

The 20 possible scores are the integers 60 through 79. Of those, 61, 67, 71, 73 and 79 are prime, and 62, 65, 69, 74 and 77 are semiprime.

Annie scores must be four of the five primes. Bonnie's scores must be four of the five semiprimes. Because the primes add up to 351, and the semiprimes add up to 347, the unused prime must be four more than the unused semiprime. Therefore, the numbers not used are 73 and 69 and each golfer's score for the four rounds was 278.

Continuing, the sum of the ten possible scores that are not prime and not semiprime is 692. The sum of the values must be 692 - 2(278) = 136. The only ways that 136 can be eliminated are to not use (60, 76), (64, 72), or (66, 70). Since there are only two odd numbers among the possible scores for Connie and Donnie, they must have been scored by the same golfer.

From this point, trial and error will show that the only possible solution is the one stated above.

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