Problem D

Arranging Tiles 2

Hugo has a box of tiles, each with an integer from \(1\) to \(9\) on it. Each integer appears on at least six tiles. Hugo creates larger numbers by placing tiles side by side. For example, using the tiles \(3\) and \(7\), Hugo can create the \(2\)-digit number \(37\) or \(73\).

Using six of his tiles, Hugo forms two \(3\)-digit numbers that add to \(1234\). He then records the sum of the digits on the six tiles. How many different possible sums are there?