  We present an algorithm analogous to the sieve of Eratosthenes that produces
the list of twin primes. Next, we count the number of twin primes resulting
from the construction with two different heuristic arguments. The first method
is essentially the same as the one in Hardy and Wright. However, the second
method is novel. It results in the same asymptotic formula but it uses a
simpler correction factor than theirs. Though we have no theory for the
accuracy of our estimates, we compute them both without and with the correction
factor and they turn out to be close to the actual counts up to 8009.
