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The first answer I got were 8,7,13,10,6,12 and 9. The second answer I got was 6,9,11,12,4,14 and 7.
First, I found all the numbers that added up to 15. They were 6+9, 7+8, 1+14, 2+13, 4+11 and 5+10.
Second, I tried the first sum in the list, 9+6. 9+6 does not work because the number 9 repeats in the 4th card as it has to equal 23. I swapped the numbers, so it is now 6+9. 6 and 9 equals fifteen, 9 and 11 equals 20, 11 and 12 equals 23, 12 and 4 equals 16, 4 and 14 equals 18 and 14 and 7 equals 21, so this is a solution.
Next, I tried 7+8. 7 and 8 does not work because number 7 appears on the last card as well. I swapped the numbers, so it is now 8+7. 8 and 7 is 15, 7 and 13 is 20, 13 and 10 is 23, 10 and 6 is 16, 6 and 12 is 18 and 12 and 9 is 21, so this is the other solution.
Then, I tried 1+14. 1 and 14 does not work because 6 and 17 equals 23, but 17 is above 15 and the cards are only numbered from 1-15. I switched the numbers, but it still didn't work because 1 and 19 will make 20, but 19 is too big.
After, I tried 2 and 13. 2 and 13 does not work because 7 and 16 equals 23, but 16 is over 15. 13 and 2 does not work either because 2 and 18 makes 20, but 18 is too big.
Next, I tried 3 and 12. 3 and 12 does not work because 1 and 17 makes 18, but 17 is too big. 12 and 3 does not work either because 3 and 17 makes 20 but again,17 is too big.
Then I did 4+11. 4 and 11 does not work because 2 and 16 equals 18, but 16 is too big. 11 and 4 does not work either because 4 and 16 equals 20 but 16 is too big.
After, I tried 5 and 10. 5 and 10 does not work because 10 repeats on the 4th card. 10 and 5 does not work either because 8 and 8 equals 16, but number 8 repeats, so there are only 2 solutions.