Hello, Chris11!

There is no formula for this problem (that I know of).

I had to reason it out . . .

A set with only a finite number of integers as members,

has the property that the average of any 3 of its members isan integer.not

What is the maximum number of elements that may have?

There are three types of integers:

. . (1) A multiple of 3: .

. . (2) One more than a multiple of 3: .

. . (3) One less than a multiple of 3: .

We musthave 3 of type (1), or 3 of type (2), or 3 of type (3).not

. . Their averages would be integers.

We can haveat most twoof one type.

We musthave one of each type.not

. . Their average would be an integer.

We can haveat most twotypes of numbers.

Hence, we can have (at most) two each of two types of numbers.

Therefore, set can have a maximum of 4 elements.