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:

with {3, 9} to fill in the unit digit

with {1, 3, 7} to fill in the unit digit

with {1, 7} to fill in the unit digit

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 .

**Path 2**: 23, 47, 61, 89 which sum to .

**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 .

**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 .

**Answer**: 220.

Nice post!

For all those that are improving their math skills, I think that it is important to note that since you could only use 1,3,7, and 9 in the ones-digit and 2,4,6, and 8 in the tens-digit, no matter how you formed the prime numbers, the sum would be the same. Of course, this would be the case even if you just stated that the numbers had to be odd (instead of prime).