The top dice cannot be 1 because to make 1 you need a 1 and a 0.
Starting with 2, the opposite number is 5 and then you can have 3,6 and not 4 because if you have 1 then the 1 on the first carriage that means the second carriage can't use 1. There are 3 possibilities.
Next is 3 and to make 3 you need 1 and 2. It has no possibilities because if the first carriage connects with a 5 then the second connects to the third.
Time to do 4 now, to make 4 you need 1+3 or 2+2 so the join is 1 then it joins to the 6 and 2 on the top. The joining carriage can also be 5,6 and 2 ; it can be 2 or 5 because it joins to a 2 or 5 and the top is 1 or 3 then the next can also be 1 or 3. The final joining carriage is 6. Then it joins to another 6. Then the top is 2, then on the next one there is 2 on the top. There are 6 possibilities.
Time to move onto 5 and to make 5 you need 1 and 4 or 2 and 3. Underneath 5 is 2 and we will start with a joining carriage of 1. The top can be 2 or 3 and then the next one can be 2 or 3. The next number is 3 and it doesn't work.
So we have found 9 possibilities. We really enjoyed investigating this.