Ooh, I've never heard of this one before! Let's see what I can come up with.

Let's see. 11 apples. Each of the four had at least one. That means the most anyone could have had is 8.

There are 7 apples unaccounted for. So, if someone had 5 (the first one, plus 4 of the other 7), then no one else could have had more than them (they could have had the first one, plus a maximum of 3 of the other 7). Therefore no one would have to ask anyone else if they'd had more.

Aju asked someone, so he had 4 or fewer apples. Therefore eat ate either 1, 2, 3, or 4 apples.

Baju asked too, so he had 4 or fewer apples. However, he knew he didn't have the same number or fewer, so he had either 2, 3, or 4.

Same thing with Caju, too, except he knew Baju didn't have only 1, so Caju must have eaten either 3 or 4 apples. But then, he asked Daju if he'd had more, so Caju couldn't have had 4 (because he knew no one would have had more than he did -- Aju had at least 1, Baju at least 2, and if he'd had 4, then the most Daju could have had would have been 4, too. Therefore, Caju must have had 3 apples!

And then, since Caju didn't know if that was more than Baju had eaten...uh...then that means that.... Ummmm....

Hmm.

Okay, I need think this through some more.

Baju had at least 2 apples. After Caju asked, "Did you eat more apples than me DAJU?" then Baju knew Caju had eaten 3 apples. Would that let him know if he'd eaten more apples than Aju? Only if Baju had eaten 4 apples. Hmm. I don't see how this helps me.

Hmmmmmmm.

IF Daju had eaten 4 apples, and he knew Caju had eaten 3, and that Baju had eaten at least 2, then he now would know that Aju ate either 2 or 1. (If Baju ate 2, then so did Aju; if Baju ate 3, then Aju ate 1.)

IF Daju had eaten 3 apples, and he knew Caju had eaten 3, and that Baju had eaten at least 2, then he now would know that Aju ate either 3, 2, or 1. (If Baju ate 2, then Aju ate 3; if Baju ate 3, then Aju ate 2; if Baju ate 4, then Aju ate 1.)

IF Daju had eaten 2 apples, and he knew Caju had eaten 3 and that Baju had eaten at least 2, then he now would know that Aju ate either 4, 3, or 2. (If Baju ate 2, then Aju ate 4; if Baju ate 3, then Aju ate 3; if Baju ate 4, then Aju ate 2.)

IF Daju had eaten 1 apple, and he knew Caju had eaten 3 and that Baju had eaten at least 2, then he now would know that Aju ate either 4 or 3 AND that Baju are either 4 or 3. (If Baju ate 4, then Aju ate 3; if Baju ate 3, then Aju ate 4.)

Guess I need to think about it some more.