How many three-digit numbers ABC (with hundreds digit A, tens digit B, and…

A number that equals the sum of the cubes of its own digits is called an Armstrong number (also known as a narcissistic number). For a three-digit number written as ABC — hundreds digit A, tens digit B, units digit C — it is an Armstrong number exactly when A3 + B3 + C3 = 100A + 10B + C.

Checking every three-digit number (100 to 999) against this condition is the systematic way to count how many satisfy it. Direct substitution confirms the following digit-combinations:

1. 1, 5, 3 → 13 + 53 + 33 = 1 + 125 + 27 = 153, which is exactly the number formed by these digits, so 153 is an Armstrong number.

2. 3, 7, 0 → 33 + 73 + 03 = 27 + 343 + 0 = 370, matching the number 370.

3. 3, 7, 1 → 33 + 73 + 13 = 27 + 343 + 1 = 371, matching the number 371.

4. 4, 0, 7 → 43 + 03 + 73 = 64 + 0 + 343 = 407, matching the number 407.

Extending the same substitution check to every remaining three-digit number from 100 to 999 shows no other digit-combination satisfies A3 + B3 + C3 = ABC — each of the four numbers above is independently confirmed by direct substitution, and no fifth number passes the check.

So exactly four three-digit numbers satisfy the given condition.