Hints:

1. Key Idea: You only need to choose which column each digit goes in—not the exact number order.
   For example, 123 + 456 + 789 is functionally the same as 159 + 423 + 786 because it’s just the same digits rearranged into columns.

2. Step-by-Step Logic:

   • The digits 1 through 9 add up to 45.

   • You are trying to split these into three 3-digit numbers.

   • When adding the numbers, pay close attention to the hundreds column (the leftmost).
     The carry into this column can only be 0, 1, or 2, so the digits in this column should ideally add up to 8 or 9:

     • If they add up to 8, the other digits add up to 37.

     • If they add up to 9, the rest total 36.

3. Casting Out Nines:

   Use the trick called “casting out 9s”:

   • The total sum of the three numbers must be divisible by 9.

   • That means you’re aiming for totals like 999 or 1008.
     Sums like 990 or 1017 might also work but are less likely.

Solution:

A key observation is that it doesn’t matter how we pair up digits within the ones, tens, and hundreds columns—only which digits go into each column. That simplifies our problem.

The sum of digits from 1 to 9 is 45.
Using a number theory trick called casting out nines, we know the final sum must be divisible by 9. So our target is 999, with 1008 or 990 as close alternatives.

Case 1: Hundreds Column = 8

This can happen, for example, with digits 1 + 2 + 5 or 1 + 3 + 4. That leaves 37 for the tens and ones digits.

We now want the tens and ones digits to sum to numbers like:

• Tens = 18, Ones = 19 → Total = 999

• Tens = 19, Ones = 18 → Total = 1008

• Tens = 20, Ones = 17 → Total = 1017

The best case is 999, with:

• Hundreds digits summing to 8

• Tens digits summing to 18

• Ones digits summing to 19

Sample answers:

• 152 + 368 + 479 = 999

• 194 + 237 + 568 = 999

Case 2: Hundreds Column = 9

This seems promising at first—but after placing three digits in the hundreds column, the remaining digits can’t make tens and ones columns that add up to the needed values to reach 999 or better. So this case doesn’t work.

Final Answer:

The best possible total is 999, using a smart allocation of digits by column sum.