Box Progression Part 3
\[ \large \square + \square\square+\square\square\square+\square\square\square\square \]
You are given that the numbers \(0,1,2,\ldots,9\) are to be filled in the square boxes as shown above (without repetition) such that it represent a sum of a 1-digit, 2-digit, 3-digit, and 4-digit number.
Find total number of possible arrangements of these nine numbers such that the sum of these four numbers is minimized.
Details and Assumptions:
This is an arithmetic puzzle, where \( 1 \square \) would represent the 2-digit number 19 if \( \square = 9 \). It does not represent the algebraic expression \( 1 \times \square \).