As pointed out to me by Tianyu (Tierry) Wang from Beihang University, there is a solution where the questions are very easy, but there are four of them.
You first ask to all three Oracles: "Does DA mean Yes?"
Independently of the meaning of DA, True answers DA and False answers BAL. So if exactly one Oracle replies DA, it is True, while if exactly one Oracle replies BAL, it is False.
Then you fix some Oracle X between the two which are still unknown to you. And you ask to the Oracle which you have identified "Would you answer DA when asked if X is Random?"
The answer DA means that X is Random, while the answer BAL means that X is not Random. To see why this is the case, try out all possibilities (asking True or False, the answer being DA or BAL, DA meaning Yes or No).
Notice that True and False give the same answer because False is wrongly describing the answer which it would give, so it presents the answer that True would give.
For example if you ask a liar: "Would you say Yes to 1+1=2 ?" The liar would of course reply No to "1+1=2 ?", so the correct answer to your question would be No, and the liar answers Yes. And if you ask the liar instead: "Would you say Yes to 1+1=3 ?" The liar would of course reply Yes to "1+1=3 ?", so the correct answer to your question would be Yes, and the liar answers No.
As pointed out to me by Tianyu (Tierry) Wang from Beihang University, there is a solution where the questions are very easy, but there are four of them.
You first ask to all three Oracles: "Does DA mean Yes?"
Independently of the meaning of DA, True answers DA and False answers BAL. So if exactly one Oracle replies DA, it is True, while if exactly one Oracle replies BAL, it is False.
Then you fix some Oracle X between the two which are still unknown to you. And you ask to the Oracle which you have identified "Would you answer DA when asked if X is Random?"
The answer DA means that X is Random, while the answer BAL means that X is not Random. To see why this is the case, try out all possibilities (asking True or False, the answer being DA or BAL, DA meaning Yes or No).
Notice that True and False give the same answer because False is wrongly describing the answer which it would give, so it presents the answer that True would give.
For example if you ask a liar: "Would you say Yes to 1+1=2 ?" The liar would of course reply No to "1+1=2 ?", so the correct answer to your question would be No, and the liar answers Yes. And if you ask the liar instead: "Would you say Yes to 1+1=3 ?" The liar would of course reply Yes to "1+1=3 ?", so the correct answer to your question would be Yes, and the liar answers No.