A Little Weird Competition

Logic Level 4

A, B, C, D, E and F took part in a competition. In this competition, the contestants were either liars either semi-liars. Liars always tell false statements, and semi-liars alternate true and false statements. After the competition has ended each of them classed a different position. Since you haven't seen the competition, you ask each of them what the final results were. From the 36 affirmations below (that is 6 made by each contestant) try to find what place classed each. Each contestant gives it's version of the positions (where each considers himself the winner) in the order from the first place on.

(A) A C D E B F
(B) B D F E C A
(C) C D E F A B
(D) D E F B A C
(E) E B A D F C
(F) F C B A E D

Write your answer by concatenating the values from 1 to 6 for the places each contestant occupied in the order A to F. As an explicit example if player A occupied 1, B occupied 5, C occupied 4, D occupied 3, E occupied 2 and F occupied 6, then your answer in the order from A to F should be 154326.

The answer is 154362.

2 solutions

Abin Das
May 15, 2016

First let us try to find out who comes first.Clearly only one of them says truth regarding the first place.Hence we have to find the semi liars among them (Clearly atleast one semi liar is present since atleast one of them is right about first place).

Consider the semi liar who came first.Clearly the positions 1,3 and 5 said by him will be right and no other semi liar will tell truth about positions 1,3 and 5 since everybody says different in 1.

Hence consider A. None of the 1,3 or 5 positions mentioned by A is not repeated by anyone else. Hence he might be the winner and a semi liar.

Next consider B.The 3rd position mentioned by him is repeated by D as his 3rd. Hence both B and D are not winners but might be a semi liar saying truth about 2,4,6 positions.

Repeating the procedure with all the six, we can find that only A,E and F can be the winners.

Now let us try to find the semi liars .Take out only the 1,3,5 positions mentioned by everybody . It can be noted that none of the six sets taken out contains the exact participants ( even with their places shuffled ) . Also none of the six sets is the complement of another ( i.e the complement of A,B,C being D,E,F ). If there were atleast two semi liars then then the alternate places when taken out ,as we did , should contain the exact participants or their complement in a shuffled manner. Hence it can be inferred that there is only one semi liar. Since the semi liar tells truth about first position, the possible semi liars are A,E and F while B,C and D are liars.

Hence the positions said by B,C and D are false and the 2,4,6 positions by A,D,F are also false .Thus making a list of all the possible positions of each player using the data of B,C and D and then comparing it with the possible 2,4,6 positions using data of A,D,F ( by the fact that the 2nd positioned player will be either in the 4th or 6th position ) and then filling up the missing places we can see that A is the only semi liar and the required positioning is AFDCBE.

I wanted to write you a more synthetic way of arriving at the answer but if you'll want I'll leave that to you. For the first part can you find a better procedure for deducing who could have been the winner ?

A A - 5 years, 1 month ago

Sorry the way I arrived at the answer is a bit long and tedious. You might be able to write a better solution and I would like you to do the same.

The part in which the winners were deduced is actually a simple task but my explanation might have made it confusing. The first thing to note here is that there is atleast one semi liar and only one semi liar who tells the truth about the positions 1,3 and 5, as I had mentioned earlier. I hope that this didn't make any confusion.

The next thing I said is that, since the winner will be telling the truth about the positions 1,3,5 and since no other semi liars of the pattern is present, no one else can repeat what the winner said about positions 1,3,5. In the case of A,E,F the positions 1,3,5 mentioned by them is not repeated by anyone else. But in case of B, the 3rd position is repeated by D and in case of C,the 5th position is repeated by D . Hence B,C,D cannot be the winners.

Hope I am clear about it this time.

Abin Das - 5 years, 1 month ago

Haha , I understood your solution since the reasoning is pretty much the same as mine but indeed it wasn't stated pretty well though in your last comment is better written. Maybe I will write a solution since you asked me to. What I meant about deducing who the winners are is that you don't have to check them like that. Another way (which I think is easier) is to look on the columns where the semi liar who won tell the truth and deduce the liars by the repeating names.

A A - 5 years, 1 month ago

@A A – Can you add your solution? I've been wanting to feature this problem.

Calvin Lin Staff - 5 years ago

@Calvin Lin – Haha , I suppose I should add solutions to all the problems which I post then. Ok then I suppose then that I will write a solution to this problem (though is pretty easy) as well. Abid Das's solution is good as well anyways and some members might complain about how I will write my solution to this problem but as long as it will be clear and I'll try a lot to make it clear even if it is long anyone interested in reading it should understand it. Anyways , I will just have to ask you just to wait a little and I will write the solution (no more than 1 day) and thanks for your motivating interest in this type of problems (and in the problems which I post) and also I'm sorry I have made you to wait that long since if I would have known anyone is very interested in it I would post it faster.

A A - 5 years ago

@A A – Writing solutions also help you improve on your presentation techniques and communication skills. As you are aware, you have a convoluted way of expressing yourself, and it's equally important to clearly and directly explain yourself to others, as it is to come up with the proof.

Calvin Lin Staff - 5 years ago

@Calvin Lin – I completely agree with you. Also writing a solution deepens the understanding as the entire reasoning gets a form and therefore helps reflexive understanding.

A A - 5 years ago

@Calvin Lin – Anyways , the source of that "convoluted way of expressing" myself has to do with the order in which I put what determines what and present it. I think this order makes things not clear because it is not seen that directly what follows from what. In a natural reasoning this comes immediately because you think it and therefore you can see what made you arrive at that conclusion and what place it has in the overall reasoning of it and therefore the understanding of such things is pretty much intuitive and immediate. However I tend to reflect on it and not just present the reasoning but also understand how I arrived at it , this being a reflexive understanding which has indeed the problem if it is not done properly of getting things in a confused and pretty much lack of the order which you see in the natural reasoning of it.

A A - 5 years ago

A A
May 26, 2016

For the purposes of this solution take the statements as forming a table where each of the first , second and so on statements of the contestants will stand in the same column and denote this columns with the roman numerals. Also for clarity I will refer to the semi-liars who tell truth in columns I , III and V as even semi-liars and the other semi-liars who tell truth in columns II , IV and VI as odd semi-liars.

It is easy to start by firstly observing that , because in column I (corresponding to the first statements of the contestants) all names that are mentioned are different , as there is only one contestant who won , just one of the statements of column I is true meaning therefore that just one contestant told the truth in the first statement while the rest are lying therefore being right to conclude that from all contestants there is only one semi-liar who tells the truth in statements I , III and V and , judging only from the fact that the rest lie in their first statement , the rest can each taken separately be either liars or even semi-liars. Because in columns III and V there are some statements which repeat (namely the statements of B , D in III and C and D in V) and there is already known that there is only one odd semi-liar all the repeating statements can't be said by the person who tells the truth and therefore contestants B , D and C can't be the odd semi-liar remaining that just A , E and F can be therefore.

To understand more clearly what are the other contestants the next step which comes naturally would also be to analyze columns II , IV and VI in their relations which covers relevant information about the potential semi-liars. Therefore looking at those columns remark that some of the statements coincide namely the statements of A , F in column II , also the statements of B , C in II , of A , B in IV and of D , E in VI. If in this columns there would be some pairs who tell the truth and therefore some even semi-liars they should tell the same thing in all of their even statements because they must tell truth and truth being just one would imply that they tell the same and therefore would imply that for the configuration all columns II , IV and VI there would be statements which coincide in the very same positions which by looking at the configuration of the same names in the columns doesn't happen. This enables further to deduce that all contestants from the pairs which tell somewhere in their even statements the same thing (A ,F) , (B , C) , (A , B) , (D ,E) must be liars and because all six of them appear in at least one of the pairs there not being possible to be a semi-liar not saying the same thing somewhere they all lie in even columns.

Now , since it is known that all contestants lie in columns II , IV and VI this means that all the persons mentioned for who won the II , IV and VI place don't occupy any of this places and because in columns IV and VI there are mentioned 5 out of 6 contestants it means that this enables to deduce who occupies places IV and VI this person being the one who is not mentioned. Looking at the columns C occupies IV and E occupies VI place.

Further. Using the new information found out this will help to deduce who out of the 3 possible odd semi-liars this being A , E and F is the actual semi-liar and the winner. Firstly , it is clear that since E occupies VI E can't occupy also place I so he can be eliminated and there will remain just A and F. Looking at statements I , III and V of A and F observe that just F contradicts what is already known saying that E is on place V. This implies as such that F is not the winner and there remains just A which is the odd one and which will be on the first place.

Until now it has been deduced that C IV , E VI and A I. Since it is known that A tells the truth in his odd statements it is sufficient to check them and what is found out is that D occupies III and B occupies V. Until now it was found out that A I , B V , C IV , D III , E VI. Since there remains only one contestant and one place which are not said namely F and II this means of course that F is on place II. The final order therefore is A I , B V , C IV , D III , E VI , F II therefore the answer being 154362. And as a simple final observation it can be said that a general analysis for any possible distribution of names and maybe also for any number of contestants , seeing what is possible to deduce and what not from such given configurations by an understanding of their logical structure would be more interesting anyways.

A Little Weird Competition

The answer is 154362.

2 solutions

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