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Then it seems the only way to satisfy all of the requirements you've been given is to add an i++ statement to be executed before the continue statement, like:
if (i == 3)
{
i++;
continue;
}...

Are you interpreting the instructions in post #3 as requiring i to be initialized to 1, or do you have other instructions that specifically say i must be initialized to 1? If the former, I don't...

I think that means you haven't figured it out. The path to a solution is understanding or finding the answer to the question I asked. You seem to have figured it out, at least you have experienced...

Okay, the requirement wasn't clear.

In either of your original while( true ) loops, when will the index, i, be incremented?

One wonders why you just didn't program it as:
while( i < 6 )
{
if (i != 3)
{
System.out.println(i);
}
i++;
}