What is the largest three-digit number with the property that the number is equal to the sum of its hundreds digit, the square of its tens digit and the cube of its units digit?

my little BASIC program gave the following:

Chipmunk BASIC v3.6.5(b3)

4 possible cases:

135
175
518
598


testing for 518

518 ---> 5 + 1^2 + 8^3 = 5+1+512 = 518
YUP!

I ran the following little program

Chipmunk BASIC v3.6.5(b3)
>10 for a = 1 to 9
>20 for b = 0 to 9
>30 for c =0 to 9
>40 if(100*a+10*b+c) = a+b*b+c*c*c then print a,b,c
>50 next c
>60 next b
>70 next a
>run
1 3 5
1 7 5
5 1 8
5 9 8
>

To find the largest three-digit number with the given property, we can start by listing all possible three-digit numbers and checking if they satisfy the equation.

Let's consider numbers from 100 to 999.

For each number, we can extract the hundreds, tens, and units digits.

For example, for the number 381:
- Hundreds digit (H) = 3
- Tens digit (T) = 8
- Units digit (U) = 1

Now, we check if the number satisfies the equation H + T^2 + U^3 = number.

For 381, H + T^2 + U^3 = 3 + 8^2 + 1^3 = 3 + 64 + 1 = 68, which is not equal to 381.

Using this method, we can check all three-digit numbers to find the largest one that satisfies the equation. Let's go through the numbers systematically:

100: 1 + 0^2 + 0^3 = 1, not equal to 100
101: 1 + 0^2 + 1^3 = 2, not equal to 101
102: 1 + 0^2 + 2^3 = 9, not equal to 102
...
998: 9 + 9^2 + 8^3 = 881, not equal to 998
999: 9 + 9^2 + 9^3 = 1080, not equal to 999

None of the numbers from 100 to 999 satisfy the given property. Therefore, there is no three-digit number that fulfills the condition.

To find the largest three-digit number with the given property, we need to follow these steps:

1. Start by considering the largest possible hundreds digit, which is 9.
2. Since we want the number to be equal to the sum of its hundreds digit, the square of its tens digit, and the cube of its units digit, we can set up the equation:
Number = hundreds digit + (tens digit)^2 + (units digit)^3
Number = 9 + (tens digit)^2 + (units digit)^3
3. The tens digit can be any number from 0 to 9, as we are looking for the largest three-digit number.
4. Similarly, the units digit can also be any number from 0 to 9.
5. We will loop through all possible combinations of tens and units digits, substituting them into the equation and checking if the resulting number matches the given conditions.
6. Starting with the largest values for the tens and units digits, we check if the number satisfies the equation.
7. If it does, we have found the largest three-digit number with the given property. If not, we continue looping through smaller digits until we find a suitable number.
8. Once we find the largest possible number, the search can be stopped.

Using this method, we can systematically check all combinations and find the largest three-digit number that satisfies the condition.