Q: How many three-digit numbers are composed of three distinct digits such that one digit is the average of the other two?

A:  112 numbers
There are four numbers that end with zero that satisfy this condition: 120 240 360 and 480.   Each of these numbers can only be rearranged four ways (as a leading zero is not allowed) making 16 combinations.

Each of the numbers below can be arranged into 6 distinct numbers.  For example, 213 can be 123 132 213 231 312 321.  Below are 16 numbers and each can be rearranged in 6 ways.  Therefore there are 96 three-digit numbers that satisfy this condition that do not have a zero in it.

The total is 96 + 16 = 112.

213

315

324

417

426

435

519

528

537

546

639

648

657

759

768

897

Using perfect squares:
This is super tricky as some of the smaller perfect squares that are 3 digits repeat digits (like 121, 144, and 225).  If we try 169 (whose square root is 13), then many of the two digit perfect squares cannot be used like 81, 64, 49, and 36).

For example, sqrt (169) – sqrt(25) – sqrt(4) would give 6.

If we bump up to 256 (whose square root is 16), then we can use 81 for the two digit and 9 for the single digit making it 16 – 9 – 3 = 4.

Using non-perfect squares:
Sqrt(145) – sqrt(92) – sqrt(6) = 0.00044178938368

Credit and thanks to openmiddle  (if you use these in the classroom, try to modify the instructions to make them less Google-able)

Q: How many three digit perfect squares are the sum of eighteen consecutive positive integers?

A: There are 3 three digit perfect square that are the sum of eighteen consecutive positive integers: 225, 441 and 729.

If there are 18 numbers in a row, there will be 9 pairs of numbers that will add up to the same sum.  Therefore we only need to check three digit perfect squares that are divisible by 9.  Another condition is that the two middle numbers in the list have to add up to that perfect square divided by 9.

144 (12^2) is even and divided by 9 = 16 each pair would have to add up to 16.  There are no two consecutive numbers that add to 16.

225 (15^2) divided by 9 is 25 so the two middle numbers are 12 and 13.
4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 has 12 and 13 for the middle numbers and every pair going outwards 11 and 14  10 and 15 etc adds up to 25.  This list would make 9 pairs of 25 or 225.

324 (18^2) is also even like 144.  324 divided by 9 is 36 and there are no two consecutive integers that add to 36.

441 (21^2) divided by 9 is 49 so the two middle numbers are 24 and 25.
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33

576 (24^2) is also even.  576 divided by 9 is 64 and there are no two consecutive integers that add to 64.

729 (27^2) divided by 9 is 81 so the middle two numbers are 40 and 41.
32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49

900 (30^2) is also even.  900 divided by 9 is 100 and there are no two consecutive integers that add to 100.

Notice each number divisible by 9 is a multiple of 3 squared.  The next multiple of 3 is 33 and 33^2 >1000.  Therefore, the only three numbers are:

225, 441 and 729.