edit 2023: The guide below is still valid for how to have the best chance of solving, but you might find this guide easier to follow: https://www.torn.com/forums.php#/p=threads&f=61&t=16254610&b=0&a=0&start=0
The section covering the method by BiggestD matches the one I describe below.

Ok, with CT coming up soon, I thought I'd share the method used to solve these chests. To many these first seem impossible, but rest assured, the combo chest mechanic does give you hints. You have to approach it correctly as a logic puzzle and there is still a little luck involved.

Counter to what you might first think, you don't want to get a number in the right spot early. The best thing to happen is to be able to continue to use all three slots to find/eliminate numbers of the combination on your first few unlocks.

Combination Chest Color indicators:
Red - wrong number, not in combination anywhere
Green - correct number, correct location
Orange (with circle of dots) - correct number, wrong location

Finding an exact number position (green) doesn't lock that number in position now, meaning you can still test numbers in that slot. (It didn't allow you to change it in the past)
This means that if you do find early numbers, you can still spend your first three turns trying 123, 456, 789.

There might be situations where you still don't have a lock after the first two tries and need to test one of your early numbers (orange), so try three might be something like (1-6)78, with (1-6) being in the position you need to test. My method and other folks examples in this thread still apply.

Here is an example of finding position on an early number (orange).

I went ahead and rotated the numbers to 4/5/6 before clicking my first unlock just to show you can start anywhere in your search. (fastest would just be to click unlock on 1/2/3 and get your first bit of info, but anyway...)


note that the 4 and 5 are red background, this means that they are not in the combination at all. The 6 however has an orange background and a circle of dots. This means the 6 IS in the combination, but in a different location. We have useful information, but we don't want to try to put the 6 in place yet. we want to search for the other two numbers and want to use as many slots as possible to find them. so next up are 7/8/9



note that they are all red background. None of those numbers are in our combination. That leaves 1/2/3 and two attempts. We know 6 is in one of the first two positions and we might need both attempts to place it, so we start now with 6 in the first position. Because we know that two numbers out of 1/2/3 are going to work on this turn, we are guaranteed at least one correct number and if it's not in the correct location, we will know it's the other location that is "not" the correct location for the 6. In this case though, the 6 and the 2 are in the correct position:


Now we know the only thing left is the 3 and the only position left is middle. We are ready to open it with our last attempt!