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First of all, I had to find two numbers which added to 15. I chose 6 & 9 because I didn’t want to use 10 & 5, the obvious couple.
I then had to find the number that matched 9 to make 20. This had to be 11.
Now I had to think of the number which you add to 11 to make 23. This turned out to be 11’s neighbour, 12.
Now, the number in question was the number you needed to add to 12 to make 16. This turned out to be 4.
I now have to make 18 with as one of my cards. I needed the number 14.
Last of all, it was the extra number I needed to make 21 with 14 already at hand. This number is 7.
My solution: 6, 9, 11, 12, 4, 14, 7.