On Friday I set this puzzle….

Last night seven of my friends went to a local restaurant. They each handed over their coat as they entered. At the end of the evening the staff returned the coats, but they were totally incompetent and randomly handed out the coats to my friends. What is the probability that exactly six of my friends received their own coat?

If you have not tried to solve it, have a go now. For everyone else the answer is after the break.

The answer is…..of course….. zero. If six of my friends have their coats back then all seven must have the right coat. Therefore it impossible for exactly six of them to have the correct coat.

Did you solve it?

I have produced an ebook containing 101 of the previous Friday Puzzles! It is called PUZZLED and is available for the Kindle (UK here and USA here) and on the iBookstore (UK here in the USA here). You can try 101 of the puzzles for free here.

1. Eddie says:

What happened to the 7th person?

1. Skipped out without paying.

2. Nick says:

I think he meant to say that if 6 have the right coats, the 7th must also.

3. I got the right answer, but I don’t understand the explanation with five and six. I would have explained it with six and seven…

1. M says:

Yes, Richard made an error there… And later on this day he’ll change it and nobody who reads it after that understands these comments. So for those, this is what it says now: “If five of my friends have their coats back then all six must have the right coat. Therefore it impossible for exactly five of them to have the correct coat.”

I suspect that you are you one of the staff at the restaurant Richard

5. Chris says:

The staff are so disorganized they lost a guest halfway through the puzzle. That said, it took the time it took to read it to figure it out.

6. Letitia says:

The 7th wasn’t a real friend :). I calculated the probability of 7 people getting their coats. Should have known it wasn’t maths…

7. “The answer is…..of course….. zero. If five of my friends have their coats back then all six must have the right coat. Therefore it impossible for exactly five of them to have the correct coat.”

Clearly Richard’s made a mistake here and meant

“”The answer is…..of course….. zero. If six of my friends have their coats back then all seven must have the right coat. Therefore it impossible for exactly six of them to have the correct coat.”

However, there is another answer. The wording of the question precludes the “totally incompetent” staff handing coats from a different set of customers to the famous seven. However, it does not disallow for the possibility that they handed more than one coat any one member of the party. It is therefore possible that 6 of the friends have the right coat with one having a superfluous extra. The possibility of that occurring is 6/(7x7x7x7x7x7x7) or rather less than one chance in 137,000. (That’s assuming any one coat is allocated randomly to any of the 7 friends and any one might get any number of coats).

8. Skyisland says:

This is a real puzzle answer…. 😊. But is funny all people thinking…. I think is Zero in all cases 7-6, 6-5, 5-4… No?…😥

1. Dave says:

Yes, exactly. But sadly some people feel compelled to repeatedly point out the slightest error on the part of someone else. I think they hope it’ll distract from their own inadequacies.

2. Twitty Boy says:

It’s always worked for me Dave. What do you do to distract from your own inadequacies?

9. Will says:

Zip, zero, obvious. Except this is Richard, who probably went with them and there could have been eight coats. Still he specifically says seven coats, so he probably didn’t wear one that night. Of course we can bet he stuck one of his friends with the entire bill using some clever little magic trick. Sneaky devil!

10. Aniruddh says:

Assumption: The 7 coats belonging to the 7 friends were randomly returned to the 7 friends.
Otherwise it is possible that 6/7 got correct coats. Probability, though, cannot be calculated as some of the variables are missing.
But I get the point about degrees of freedom.

11. mittfh says:

One sneaky alternative explanation: the staff lost the seventh coat, or someone slipped into the wardrobe and stole it while the friends were eating dinner…

Of course, both would be unquantifiable in terms of probability unless you happened to know the frequency of staff mislaying coats or the local crime rate (particularly for coat thieves)…

12. Lazy T says:

Easy x 7, I think the real puzzle was inventing a new one around the scenario. What if all the coats were identical bar their odour, could they be sorted out in the miasma of cooking smells?

13. Simple rule of thumb: If any math problem can be solved by me in less than ten minutes, it’s too easy.

14. Dharmaruci says:

i remain confused. why is it not possible that the seventh recieved his own coat in error? that would fit the puzzle conditions.

1. Niva says:

Look, if there were 7 coats belonging to 7 people and 6 of those got their own coats, then the 7th coat had to go to the 7th person. Is this not obvious? If it isn’t, create 7 slips of paper of different colors. Cut them all in half. Now mix them up. Then take 6 of the halves and match them to the 6 halves with the same colors. Well, what do you see for the remaining two slips?

2. Dharmaruci says:

i thank you niva for trying to explain to me. but i still do not see it. by your reasoning if there were six diners then it is impossible to get five coats right. surely therefore by Mathematica Induction we will reason down to one coat and one diner. and that one coat must 100% be right, although Induction would say it was not.. so therefore with 7 it must be 1/6 = approx 20%

3. spiderabc1 says:

Ha! My final answer was 4. Four of the 6 will get the correct coat. Make out of that what you will. #mathguess

4. If there are seven people and seven coats, and six of the people correctly receive their own coats, then whose coat will the seventh person receive? There’s only one coat left–his own–so it is impossible for him to be the ONLY person to receive the wrong coat.

15. Kristian says:

Oh, I thought it was 1:720. I didn’t think the 7th coat being correctly distributed would be a violation. But I see why it is.

16. Indigo May Roe says:

So obviouse-but I didn’t get it right-your trixy language tripped me up… Good one though!

17. The other Matt says:

I got 1*2*3*4*5*6*7=5040th chance. My thought was also that also the seventh got automatically the right one, but i didnt read the “…exactly six…” in the question.

1. droz says:

I got the same answer, assuming that the 7th person died before he could receive his coat 🙂

18. The answer depends on the historical check-to-coat matching ratio of the clerk. If he or she gets it wrong every 7th time, then there’s a 100% chance that 6 have the correct coat, and the 7th guy gets the coat of some poor slob who left without claiming his years ago back when patches on the elbow were still popular. Luckily the coat pockets contained \$23.50 in small bills and coins, a map to the New York subway system, a silver ring in the shape of a unicorn, ticket stubs from a performance of CATS in 1982, and a half-empty pack of still quite usable chewing gum.

1. Lazy T says:

That’s mine! unicorn ring, chewing gum, yipee
I wondered where I left it, but I’ve never been to NY or Cats, what a puzzle.

19. Richard states the problem above as ” At the end of the evening the staff returned the coats, but they were totally incompetent and randomly handed out the coats to my friends. What is the probability that exactly six of my friends received their own coat?”

Clearly he has omitted some sentence here specifying that six received the right coat. But only Nick seems to have noticed this.

This one was absurdly easy. Yet it seems that some didn’t get it. The real puzzle is why they didn’t. What was the mental glitch? Hard to see.

Whatever it was, it is very unexpected. The simple logic of the question seemed to leave no room for imaging anything else but the right answer at once.

Judging from the rest of the comments here some people have the most disordered brains, constantly beset by imaginings both absurd and irrelevant to the point of the question.

If this is characteristic of the population at large, no wonder the results of voting in the US are such a poor validation of democracy.

In fact on this basis I predict Romney will displace the impeccable professorial pragmatist Obama.

1. I’m not sure what you’re talking about.

The puzzle *does* include a sentence specifying that six received the right coat. You quoted it in your post! “What is the probability that exactly six of my friends received their own coat?”

Nick’s post was in response to an earlier version of Richard’s solution, since replaced with a corrected one, in which he inadvertently answered the question as if five out of six friends (instead of six out of seven) received their own coats.

20. Anonymous says:

Damit! I just read the trains puzzle thinking that was a trick and got it wrong because I had to use math (technically) so I worked this one out with math when it turned out to be a trick 😦