The Blind, the Deaf, the Mute

Logic Level 4

Four friends named A, B, C, D were challenged to play a weird "Guess-A-Word" game. The 4 contestants would be separated into 4 different chambers, where they couldn't see or hear each other unless permitted, with the following conditions:

One of the contestants would be blindfolded such that he could not see anything but could still listen and speak well. (The Blind)
One would be ear-plugged such that he could not hear the others but could still see and speak a word to others. (The Deaf)
One would be mouth-sealed such that he could not speak but could see and listen well. (The Mute)
The remaining one would not be constrained by any means at all and perceive all senses. (The Normal)

Initially, none of them knew who was which. Then the game would proceed as follows.

First, A would be secretly shown a word in text. They would then need to verbally tell B by voice. (The walls made B only would hear this. B couldn't see A even B was not blind, so A couldn't communicate using sign language or anything else though if A was deaf and B was blind, B could still hear A's word.) B would then have to say the word aloud to get a point.

Next, it's B's turn telling a new word to C, with the same rules as above. Then it's C's turn telling D, and lastly D's turn telling A.

After the game concluded, the four friends didn't get any point. They then discussed the game:

B : That was a hard game!

A : Indeed! I wonder what word you've got, C?

C : Not a chance. I've only known one word in this game. I'm going to keep it my secret.

D : Such a pity. I wish I could know your word, C.

What identities were A, B, C, D during the game? Let 1 = The Blind, 2 = The Deaf, 3 = The Mute, and 4 = The Normal; enter your answer as the identities of A, B, C, and D in order. For example, if you think A is The Blind, B is The Deaf, C is The Mute, and D is The Normal, then you should enter $1234$ as your answer.

The answer is 3142.

2 solutions

Worranat Pakornrat
Mar 31, 2017

To earn no points despite having the normal contestant within a pair, his teller must be either blind or mute, for the deaf one can still see and tell a word to him. Thereby, if the normal one is a teller with no success, his listener must be either deaf or mute, for the blind one can still listen and guess the word. Then there are $3$ possible scenarios:

As shown, only the second scenario in the middle works because if the deaf one tells the blind, he can still hear the word and can answer correctly.

As such, after knowing that they earn no points, the four friends will already know such order, as each of them knows his own status. Following this cycle, the blind can not possibly see his word and can not hear anything from the mute, so he knows no words. For the normal one, he will not know the blind's word as the blind though the normal will know his $1$ word. Then the deaf will not hear the normal's word but can see his own and will know $1$ word. Finally, the mute can hear the deaf's word and can see his own word, so he's the only one who knows $2$ words.

To answer the question, we just need to know which is which by analyzing their conversation. Since $C$ knows only one word, he can't be blind or mute: only the normal or the deaf knows one word. However, if $C$ is deaf, $D$ will be mute and can then know his word. Thus, $C$ is normal, making $D$ deaf, $A$ mute, and $B$ blind, respectively.

Even after reading your solution, it took me a while to understand what you're trying to say.

In the problem, it might help if you added "For example, if A was deaf and B was blind, then A can speak to B".

It needs to be much more obvious what "tell a word" means. IE specify that we can write it down (if we can see) or say it out (if we can speak).

Calvin Lin Staff - 4 years, 2 months ago

Is it better now?

Worranat Pakornrat - 4 years, 2 months ago

I do not see any substantive improvement. I'm still confused about

The ways of communication that are allowed
The ways of communication that are enforced by the rule description
The number of distinct words that people tell each other
How pints are scored

In more detail, the questions I have after reading are:

Must A only "shown a word in text"?
Must A only tell B via "verbally tell B through voice"?
What are the ways that B can tell C?
Why would C refuse to tell A? In particular, why does C say "I only known one word"? If the game only involves one word, then it should be "I did get a word from B". However, if A, B, C, D all have different words to pass around (which makes sense given that he only received one word), then that needs to be made clear.
How are points counted in this game? What constitutes scoring a point? If A successfully tells B, is that a point? Or must it be A tell B tell C tell D tell A, in order to score a point?

Calvin Lin Staff - 4 years, 2 months ago

I will try to answer these questions. 1. Yes
2. Yes
3. Same as A tells B 4. There are multiple questions
First question This conversation is after the game is finished so anybody can do now what they like Second question The game for any particular person involves 'two' words; first that he hears and is supposed to say aloud to gain a point, second that he sees and supposed to tell the next person. Blind will be able to hear the first word but not see the second. Deaf would not be able to hear the first word but would be able to see the second. 5. A hears a word from D and if he speaks it out loudly then he gets a point. Now A is shown a 'different' word and tells it to B and if B says it correctly B gets a point and thing goes on. I hope with this clarification now you would be able to understand the rationale of the answer

Zahid Hussain - 1 year, 10 months ago

Saya Suka
Mar 4, 2021

Firstly, the Normal can both receive all inputs (visual and auditory both) and transmit an output (speak out any received inputs), so if Normal had problems in getting a point, then those must lie within his / her partners. Either Normal's first role partner can't see the text or can't speak the work out, AND either Normal's second role partner can't hear the word or can't speak the work out.

Anyway, the requirements for a useful first role are that they can both see and speak, while the requirements for a useful second role are that they can both hear and speak. That's why we have the Deaf to be an ideal first role while the Blind to be an ideal second role along with Normal who's obviously both. Therefore, by the information that none of the pairs gained any points, we know that :
1) the Deaf cannot come directly before the Normal, and
2) the Blind cannot come directly after the Normal, and
3) the Deaf cannot come directly before the Blind (equivalent to the Blind cannot come directly after the Deaf).

Thus, Deaf can only come before Mute and Blind can only come after Mute. With this, we get a chain of { Deaf --> Mute --> Blind --> Normal --> Deaf }
Having the cyclic chain fixed, we can score them with 0 or 1 with failed or successful inputs and outputs conveyance respectively.

[ 1 ] Deaf [ 1 ] --> [ 1 ] Mute [ 0 ]
The above have these meanings :
One before Deaf : s/he can see the word in text, since they're not visually impaired.
One after Deaf : s/he can speak the word out loud, thus output successfully sent.
One before Mute : s/he can hear the word clearly spoken by the first role partner, since they're not of hearing impaired.
Zero after Mute : s/he cannot speak the word out loud, thus output failed to be sent (to the judge) and no points gained.

The turns in a full cyclic chain would be as depicted below :
[ 1 ] Deaf [ 1 ] --> [ 1 ] Mute [ 0 ]
[ 1 ] Mute [ 0 ] --> [ 0 ] Blind [ 0 ]
[ 0 ] Blind [ 0 ] --> [ 0 ] Normal [ 0 ]
[ 1 ] Normal [ 1 ] --> [ 0 ] Deaf [ 0 ]

We can infer that either Normal or Deaf could be C with having only one input available to them, but by D saying that they wished they could know what C's word was implied that D did NOT and could NOT do so during their turn of the game then. This can't be Mute who did hear Deaf, and therefore C must be Normal and D is the Deaf. Accordingly, A is Mute and B is Blind.

"Given that 1 = The Blind, 2 = The Deaf, 3 = The Mute, and 4 = The Normal; enter your answer as the identities of A, B, C, and D in order."

A : the Mute = 3
B : the Blind = 1
C : the Normal = 4
D : the Deaf = 2

Answer = 3142

The Blind, the Deaf, the Mute

The answer is 3142.

2 solutions

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