# Find Out Our Numbers (liars and truth-tellers)

You are an adventurer to a distant jungle that the news told you it was full on secrets. But, you could not find anything! So you decided to head home.

When you were almost there, a troll (you ca't see his face) tells you.

"LOLOLOLOLOLOL explorer you're bleeped-- you have fallen into our trap! I will eat you and you will suffer and die!"

"But please! I'm too smart for that! I am a genius for logic puzzles and everything--"

Then a goblin with a loud voice said, "Did someone say genius? We love logic puzzles!Okay we give you a chance to live, but first, you have to pass this challenge!

If you want to live, you have to correctly guess our names and do the following sums in our number system, and answer is decimal numerals (e.g. 420).

⨋⨈⨇ ₡ ⨊⨇⨇ =

⨈⨈⁜ ≈ ⨇⨊⨊ =

⨈⁜ ℗ ⨊ϴ =

⨈⁜⨊ϴ ֏ ⨋⨇ =

The operators shown above are add, subtract, multiply and divide, but you don't know which is which.

Also, the number system that we use is somewhere between base 8 and 60, but you don't know precisely what. And you don't know what the above digits represent either. We also have seperate signs for yes and no, but you don't know what they are yet.

And finally, we don't have floating point or negative numbers, so division rounds down, and subtraction takes the absolute value. You can only ask yes/no questions and math questions with only our digits and operators. So you can't ask, 'What is 4/2?' but you can ask, 'What is ⨇℗⨋?'

If you ask anything other than yes/no and specified math questions, we will answer by kindly bringing you to our kitchen.

If any of us can't answer a yes/no question, or divide by zero in a math question, they will leave and you can't answer them anymore.

So you know the rules and so do I, let's introduce ourselves:

My name is A and I am king of the goblins. I will tell you the truth for every y/n and math question.

B is our rebel. He will lie, and subtracts the base (you don't know what it is yet) from the answer to every math question.

C is the mericless. He says yes and no randomly in y/n questions, and add a random number between 1 and the base to a math question's answer.

D is the toggler, who starts off telling the truth, lying, truth, lying etc in y/n questions, and for math questions, he will tell you the truth on his first, add 1 to his next, add 2 next, add 3, add 4 and so on.

E is the joker. He will telepathically ask me or B randomly, and for math questions, will randomly add 1 or subtract 1.

F is the meteorologist. If it was raining on his imaginary planet last week, he shell behave like me. Else, he shall behave like B.

G is the yessayer. he always says yes, or 42 in math questions (because it is the answer to everything)

Finally, H is the nosayer. He always says no, or 0 in math questions.

(The troll anters the group of trolls, because all of them look the same, you don't know where they went)

Now, it's your turn. Ask one of us our first question.

## 1 Answer

Extremely inefficient method:

Assuming that whether or not it rained on the Meteorologist's planet a week ago remains constant the entire time, so he will either be completely truthful or completely lying.

1: Identify somebody useful (the king, rebel, or toggler).
First, ask each of them "Are you the king? Are you the king? Are you the rebel? Are you the rebel? Are you the toggler? Are you the toggler?"
Call the first answer a person gives "A", and whatever answer doesn't match that "B". A/B will differ for each goblin.
The king answers AABBBB. The rebel says AABBAA. The toggler says ABABBA. The meteorologist, yes man, and no man say AAAAAA. The merciless and joker might say anything.
At least one of the patterns AABBBB, AABBAA, or ABABBA must have only one person giving it. Whoever gives pattern can be identified.

2: Identify the king
If I identified the king in the 1, I go to step 3. If I identified the rebel in step 1, I know that A means yes and B means no, and he's going to lie. I ask "Is ___ the king?" about the other goblins until I get a no answer.
If I identified the toggler, I ask "Is ___ the king?" about the other goblins. Group the goblins asks about in odd questions and even questions separately. Whoever got the different answer is the king.

3: Identify everyone else
Whatever he answered his first question means yes. Ask him "Is ___ the ___?" about every goblin and each role.
I now know every goblin's role, and also the values for 0 and 42.

4: Learn the number system
Operators are ₡ ≈ ℗ ֏
Ask for 42₡42, 42≈42, 42℗42, 42֏42
Whichever op gives 0 is minus. Whichever gives a 1 digit number is division; the result of that is 1.
Now that I know 1, I ask for 1#1 and 1@1. Whichever op gives 1 again is times; the last is addition.
Ask for 111..111 (at least 62 ones) times itself. (If the length of the number is less than the base, it will have a nice palindromic form like 123454321; if it is longer than the base, it have a form like "..2054321" or "..207654321", ending in two, zero, all digits in decreasing order. Since I'm told the base is at most 60, the former won't happen, but if it does, just repeat with much longer numbers until the latter happens.) This will give me the base and all the digits.

At this point I know everything. And then they eat me because the one who claimed to be king was lying all along.