There are four integers between 100 and 999, inclusive, which equal the sums of the cubes of their digits. Three of them are 153, 370, and 407. For example, 153 = 1³ + 5³ + 3³. What is the fourth such integer?
Since 1 = 1³, adding it to both sides of the equation would not disturb the equality if on the non-cubes side of the equation it could replace a 0 (which of courses adds nothing to either side of the equality) in the one's place. Fortunately, the already-provided 370 allows such an addition of 1, and the answer is 371. 3³ + 7³ + 1³ = 371.