Let's say the cards played are numbered and satisfy the condition that the sum of the first two, four, six, .., sixteen cards are all multiples of 9.
Working modulo 9, we must have
Where is 9 in this list? There is no solution to with and , so we must have .
So all the integers from 1 to 17, except 9, must be among the first 16 numbers in the list: . Let's say and are "friends", as are and ,
and , ... and . Every number from 1 to 17, except for poor lonely 9, must have a friend. A pair of friends must sum to a multiple of 9.
Who are the possible friends of 1? Only 8 and 17. Let's assume 1's friend is 8. Then 17's friend must be 10 because 1 is already taken. Who can be 10's friend? Nobody is left; 8 and 17 are already taken. So 8 can't be 1's friend. It's left to the reader to show a similar contradiction arises if 1's friend is 17.
So no such arrangement is possible.