Consider the following triples:

(#1,#2,#3) slots

(#4,#5,#6) slots

...

(#34,#35,#36) slots

There are 12 on them. Say the max sum is 54. Then there total sum is bounded by (12)(54) = 648. While the total sum is 1+2+...+36 = 666. Which is an impossibility. Thus the max sum of these must be at least 55.