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I have fifteen cards numbered 1− 15. I put down seven of them on the table in a row. The numbers on the first two cards add to 15. The numbers on the second and third cards add to 20. The numbers on the third and fourth cards add to 23. The numbers on the fourth and fifth cards add to 16. The numbers on the fifth and sixth cards add to 18. The numbers on the sixth and seventh cards add to 21. What are my cards? Can you find any other solutions? How do you know you've found all the different solutions?