Two Digit Primes

The digits 1, 2, 3, 4, 6, 7, 8, and 9 are used to form four two-digit prime numbers.  If each digit is used only once, find the sum of the four two-digit prime numbers.
Source: mathcontest.olemiss.edu 11/17/2008

SOLUTION
The two-digit prime numbers are: 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 61, 67, 71, 73, 79, 83, 89, and 97.

Since each of the even digits {2, 4, 6, 8} is used only once, we form the four numbers as follows:

$2\;\_\_$ with {3, 9} to fill in the unit digit

$4\;\_\_$ with {1, 3, 7} to fill in the unit digit

$6\;\_\_$ with {1, 7} to fill in the unit digit

$8\;\_\_$ with {3, 9} to fill in the unit digit

Choose 23
If we choose 23, then the next number could be 41 or 47. We now have two paths:

Path 1: 23, 41, 67, 89 which sum to $23+41+67+89=220$.

Path 2: 23, 47, 61, 89 which sum to $23+47+61+89=220$.

Choose 29
If we choose 29, then the next number could be 41, 43, or 47. We now have three paths:

Path 1: 29, 41, 67, 83 which sum to $29+41+67+83=220$.

Path 2.1: 29, 43, 61, 8 __ . There is no solution.
Path 2.2: 29, 43, 67, 8 __. There is no solution.

Path 3: 29, 47, 61, 83 which sum to $29+47+61+83=220$.