Inspired by this question.

Two friends (Bob and Kim) go to another friend (Bill) and claim to be able to read each other's mind. Bill doesn't believe that and wants to test their ability.

"Before we begin, I must tell you, our abilities work better if it is something simple we can focus on, such as a number or a color. Let's start with a color just to warm up, shall we?" Kim says.

Bill agrees, so Bob steps away from Kim and Bill, gets blindfolded and waits. Meanwhile Bill whispers something in the ear of Kim. Kim then says: "Ok, Bob, are you ready to establish the mental link?" "Yes I am" replies Bob. "Very well then, let us establish the link."

At this point both Bob and Kim begin making beeping noises which sound like a cross between tuning a radio and an alien spaceship landing.

"The link is established," says Bob.

"Yes, the link is established," replies Kim.

"Are you thinking of a rabbit?" asks Bob.

Kim sighs. "Colors, Bob!" replies Kim. "We're doing colors!"

"Oh right." replies Bob. He thinks for a minute and then replies, "Is it blue?"

"Yes!" replies Kim.

"Lucky guess," says Bill. Let's try again. He whispers something else into Kim's ear.

Kim starts to speak, but Bill covers her mouth. "You can guess now, Bob," he says motioning for Kim to remain silent.

Clarification Hint:

Bill didn't want Kim to speak, because he was trying to trick Bob and was afraid that she would blurt out that Bill had told her a number, not a color.

Bob thinks for a minute, then says, "I'm having trouble seeing your thoughts clearly. Are you sure you are concentrating hard enough?"

"No," replies Kim and she screws up her face and glares at Bob as if trying to transmit her thoughts into his brain like a nail.

"Is Bill still in the room?" asks Bob.

"Yes he is," replies Kim.

Bob scratches his head. "Has he told you the color yet?"

"He already did that!" replies Kim.

"Oh," says Bob. Are you sure?"

Kim looks at Bill and just sighs deeply.

"All right, all right!" says Bob. "It's coming to me, it's coming to me, it's... Bill, you didn't give Kim a color at all, you gave her the number 34!"

Bill looks at Bob in shock. "How did you know that?" he asks.

How did Bob know that?

Bill decides he wants to see another demonstration, so he whispers something else in Kim's ear. Kim says nothing, so Bill finally says, "Go ahead, Bob!"

"Oh I'm sorry, says Bob. Was I supposed to be guessing again? I think our link dropped. Kim, are you ready to re-establish our link?"

"Yes," says Kim.

At this point both Bob and Kim begin making beeping noises which sound like a cross between tuning a radio and an alien spaceship landing.

"The link is established," says Bob.

"Yes, the link is established," replies Kim.

"Kim, I sense Bill is getting impatient. Do we need to step up our game?" asks Bob.

"No," replies Kim.

"Ok," replies Bob. "Are you thinking of a squirrel?"

Kim turns to Bill. "I am..." she sighs. "Sorry, Bill." She shakes her head sadly. "Give him a..." she looks into Bill's eyes. "No? No, Bob."

"Oh right," replies Bob. "We aren't doing animals this time, are we? I keep telling you we should do animals, but you are always like, 'No Bob, animals are too hard for me to focus on.' Tell you what, we will switch next time and Bill can whisper an animal in my ear and you can tell me what it is, right after after I guess that Bill told you the number one-thousand seven hundred and forty two."

"Inconceivable!" exclaims Bill. "How do you keep doing that?!"

How do they keep doing that?

"Ok my turn," says Bob. "Whisper an animal into my ear and Kim will guess it."

Bill whispers something into Bob's ear.

Bob then says: "Ok, Kim, are you ready to establish the mental link?"

"Yes I am" replies Kim. "Very well then, let us establish the link."

At this point both Bob and Kim begin making beeping noises which sound like a cross between tuning a radio and an alien spaceship landing.

"The link is established," says Kim.

"Yes, the link is established," replies Bob.

"Are you concentrating on sending me the image of the animal yet?" asks Kim.

"Yes I am," replies Bob.

"Ok," says Kim. "Are you sure?"

"Absolutely!" replies Bob.

"Ok, I am receiving the image now. Can you sense me receiving properly?" asks Kim.

"Yes, yes" says Bob.

"Have you finished sending it?" asks Kim.

"Yes I'm done." replies Bob.

Kim squints slightly, then appears to be looking at the ceiling for a moment. Then she says, "Is it a raccoon?"

Bill faints. How did Kim and Bob do it this time?

Hint:

The beeps are just something to throw Bill off. However, Bob and Kim are passing the information to each other in plain sight. The fact that Bob and Kim can't seem to remember/decide what kind of information they are going to be sending each other plays an important role in this.

Update:

The next day Bill corners Kim and Bob and demands another session. Bob and Kim agree, on the condition that Bill remain seated, so that he won't faint again. However, when they are ready to begin, Bill insists that Bob reads his mind, instead of Kim's. Reluctantly they agree, and Bob is blindfolded once again. Kim whispers something in Bill's ear.

Bill then begins imitating Kim, and says: "Ok, Bob, are you ready to establish the mental link?"

"Wait, wait wait!" says Kim.

"What?" says Bill.

"Are you sure you know how to establish the link?" asks Kim.

"I saw you do it yesterday, didn't I?" asks Bill.

"Yeah, I guess you did huh," says Kim.

"So can I start now?" asks Bill.

"I guess. Sorry to interrupt your concentration like that."

"You should be," replies Bill.

Kim nods.

"Yes I am" replies Bob.

"Very well then, let us establish the link."

At this point both Bill and Bob begin making beeping noises which sound like a cross between tuning a radio and an alien spaceship landing.

"The link is established," says Bob.

"Yes, the link is established," replies Bill. "I am sending the image now." Bill appears to be concentrating heavily.

"Sorry to interrupt, but I don't um..." begins Kim, her voice trailing off as she sees Bill glaring at her.

"I'm sorry Bill..." she says he stares at her for a few seconds.

"Are you sure you are on the same wavelength?" she looks hesitantly at Bill, until suddenly Bill's frown turns into a smile.

"You're just stalling," he announces confidently. "You know Bob can't read my mind and you're just trying to delay the obvious fact coming to light, aren't you?"

"No, Bill." Kim announces dejectedly, as she folds her arms and sits down with her back to him.

"Can we continue then?" Bill asks. Kim does not reply, so Bill continues happily. "Good. Guess away Bob."

Bob appears to be muttering something obscene under his breath.

"What's that Bob?" Bill says, taunting him. "Frickin hell you can't guess what I'm thinking, did I hear you right? Guess you can't read my mind after all!"

Bob chuckles, then replies. "I don't have to guess Bill to know that you were thinking of an elephant! And not just plain elephant, not an Asian elephant, but an elephant of the African variety!"

Bill looks suddenly stricken, but fortunately does not fall to the ground as he is already sitting.

How did Bob and Kim do it again? If you think you know the gist of what is going on, post your own mind reading session between Bob and Kim. If there are any answers that are on the right track, I will accept the closest one at the end of the day today. Otherwise, I will add another session tomorrow.

• i'm dying to know the answer.. Commented Nov 14, 2014 at 6:55
• @Geobits Sorry, it was late last night and I was feeling slaphappy and imagining Bob and Kim mimicing Kris Angel... Commented Nov 14, 2014 at 15:12
• I'm very intrigued by this question but I must admit I have no idea! Commented Nov 14, 2014 at 20:57
• Don't mind me, I'm not very good at these! :) Just hanging around waiting to see if someone else will solve it. Commented Nov 14, 2014 at 21:14
• @liebs19 It's getting quite long, but I have added another session. Commented Nov 14, 2014 at 21:34

In the first example,

the "four-legged then the answer" pattern seems to have the guesser and the sender's positions reversed from the original puzzle; is it intentionally different?

I notice that in the second example,

Kim answers "no" to Bob's initial question, then responds with a three-word phrase ("Yes he is"), a four-word phrase ("He already did that"), and a no-word phrase (sigh), and the answer turns out to be 34. It could be that the first question is to establish whether or not Bill is thinking of a color, and the number is determined by number of words in each sentence, with sigh terminating the number.

However, this seems to fall apart in the third example:

Kim answers "yes" to Bob's question about whether she's ready to begin, maybe meaning that yes it is a number again (this seems like a weak guess to me though). Kim then replies with a one-word phrase ("No"), a two-word phrase ("I am"), a two-word phrase ("Sorry, Bill"), a four(?) word phrase ("Give him a...No?") and a two-word phrase ("No, Bob.") This gets 12242 rather than 1742 though.

I don't know about the fourth one, but in the fifth (update) example:

Bob seems to be muttering "african elephant" and Bill mishears this as "frickin hell", but I don't know if that's a clue or just part of the story that wasn't obvious to me on first reading.

• With regards to your first comment, yes the sender and guesser's position is reversed. In this puzzle the receiver asks the questions, not the sender. Commented Nov 14, 2014 at 22:47
• Regarding to the muttering/mishearing, that is very definitely a big clue. Commented Nov 14, 2014 at 22:49
• Your comment on the second answer is exactly right, with the addendum that Bob wanted to make sure Bill was sticking with a color. Commented Nov 14, 2014 at 23:00
• With regard to the third example, one clue is that the system used depends on the type of info being sent. Let's just assume Bob and Kim are actually math savants. Commented Nov 14, 2014 at 23:01
• Thanks for the hints, this is a pretty fascinating puzzle! Commented Nov 14, 2014 at 23:10

Is it

the beeping noises? Probably Kim and Bob agreed on alphabet. Something like morse code and "transmit" the messages via beeping?

Or might it be some

sign language? Bill is blindfolded after all.

• I don't think so :) they are always speaking between a link and another... Commented Nov 14, 2014 at 7:12
• @Noldor130884, well they are establishing the link always after one of them knowing the answer. So, theoretically they have the information to be transmitted. Commented Nov 14, 2014 at 7:17
• Morse is what I was thinking, too, except for the time he "guessed" 34. No beeping involved. Commented Nov 14, 2014 at 7:51
• Oh, I just understood, that my second guess is wrong, because Bob and not Bill was blindfolded. Commented Nov 14, 2014 at 8:37
• Bob and Kim did agree on something before hand, but the beeping noises isn't transmitting any information. Commented Nov 14, 2014 at 20:49

They are using Morse Code when establishing the link. However, the jig was almost up when Bill decided do play dirty. Due to Kim's behaviour and Bill being a friend of both of them Bob deduced it wasn't a colour, so he hazarded a guess and went for Bill's favorite number.

• i think a guess to choose Bill's favorite number is not a reliable explanation, still waiting to know the correct answer Commented Nov 14, 2014 at 9:17
• Yes, I think so, too. But it's thew only one I could come up with. Commented Nov 14, 2014 at 9:19
• The link establishment beeps are just to throw Bill off. I'm adding a hint. Commented Nov 14, 2014 at 15:16

Kim is ventriloquist. While making 'the weird voice', he mimicked the answer. That's why when Bill keeps Kim shut up (covered his mouth), Bob can't think.