The three-digit number with a twist

Let the three-digit number be $abc$ :

$100a + 10b + c = a + b^2 + c^3$

By rearranging, we get:

$99a + b(10 - b) = (c - 1)c(c + 1)$

Seeing that when $a = b = 4, 7$ and $c = 3, 8$ , it produces solutions, substitute these values into the equation. We get:

$n = 135, 175, 518, 598$ .

Therefore $\fbox{n = 598}$ is the largest solution.

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import math

for a in range(1, 10):
    for b in range(0, 10):
        for c in range(0, 10):
            if (100*a + 10*b + c == a + b**2 + c**3):
                print (a,b,c)

Output

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2
3
4

1 3 5
1 7 5
5 1 8
5 9 8

The largest number is $\boxed{598}$