the question makes sense if "only" is replaced by "any" that is if the question is the following:

When the points scored by six basketball players during a game were arranged in order, the difference between any two consecutive totals was 9. The average of the six players' points was 36.5. How many points were scored by the highest scoring player?

let the scores of the six players be: a1, a2, a3, a4, a5 and a6, a1<a2<a3<a4<a5<a6 and a1,a2,a3,a4,a5 and a6 are non-negative integers

the difference between any two consecutive totals was 9: a2 - a1 = a3 - a2 = a4 - a3 = a5 - a4 = a6 - a5 = 9 pr we can write

a2 = a1 + 9

a3 = a2 + 9 = a1 + 9 + 9 = a1 + 2*9

a4 = a1 + 3*9

a5 = a1 + 4*9

a6 = a1 + 5*9

the average of the six players was 36.5

(a1 + a2 + a3 + a4 + a5 + a6)/6 = 36.5

(a1 + a1 + 9 + a1 + 2*9 + a1 + 3*9 + a1 + 4*9 + a1 + 5*9)/6 = 36.5 multiply both sides by 6

(5*a1 + 9(1 + 2 + 3 + 4 + 5)) = 36.5*6

5a1 + 9*15 = 219

by solving we find

a1 = 16.8

click here to see the step by step solution of the equation

since we got a solution that is not an integers follows the problem has no solution.