Tonight I am talking about the psychology of sleep and dreams at the Edinburgh International Science Festival with Caroline Watt – details here.  Hope you can make it.  If you do come along, please say hi.

On Friday I set this puzzle….

Linda is the CEO of a big company. One day she walked out of her modestly sized boardroom and said to her secretary, ‘Here is a report about the company accounts for last year. I would like you to make a copy of this report for all of the board members, plus one copy for me and one copy for my mother’. Linda’s secretary, John, jumped up and ran to the photocopier. He returned exactly 11 minutes later, after photocopying 169 sheets of paper. How many board members are there?

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

169 can only be divided by 1, 13 and 169.  Given that there must be more than 1 copy, and it is unlikely that there are over a hundred board members, John must have made 13 copies.  One went to Linda, one to her mother, and thus there are 11 members of the board.

Did you solve it? Any other solutions?

I have produced an ebook containing 101 of the previous Friday Puzzles! It is called PUZZLED and is available for theKindle (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. There are certainly 11 members of the board, but Linda, as CEO is surely a board member too so she will have received two copies. Of course if her mother was on the board too, she also would have received two copies.

1. Michael Sternberg says:

This time I have to stick to the official answer being proper and complete, as in (11 + 2) * 13 = 169 being the relevant decomposition that nets full score.

If one really wants to puzzle about Linda, make her Chair of the board, as opposed to a mere member.

2. The CEO (or MD in pre-US terminology UK business) is often the chair of the board, but the chair of the board is also a member of the board. CEO’s are also, of course, answerable to the board but they are still a member of it. In some European countries there are separate administration boards (for executive directors) and overseeing boards (for non-execs), but that does not seem relevant here.

The only answer that fits the wording is that there are 11 board members with Linda receiving two copies, one as a member of he board and the second as a personal copy.

2. Cur says:

What happened to the original?

3. simontaylor says:

My answer was 12, so I’m disappointed. The CEO is a member of the board as well, so 11 others and Linda makes 12,

1. I got that he must have made 13 copies but also concluded that there were 12 members of the board since the chief executive officer is normally a board member also

2. The CEO is always a board member. In any case, the problem says it’s her boardroom to remove any doubt.

3. M says:

It’s 11. It says “I would like you to make a copy of this report for all of the board members, plus one copy for me and one copy for my mother’”

Since it is a copy for “all of the board members, plus one copy for me” It excludes “me” from the “board members”. No matter if Linda is member, it is 11.

4. Michael Sternberg says:

Yup, pretty straightforward and clearcut.

I like the careful elimination of the edge cases 1 and n = 169. It’s all too easy to forget these when considering conspicuous prime factorizations.

5. I loved the (subconscious?) information leak regarding the number of board members: “…exactly 11 minutes later…”

6. simontaylor says:

I can only deduce that the wording implies that Linda had an EXTRA copy for herself, but why, when she also has the original?

1. Surely the original stays in the company files, so Linda would be given one of the copies for herself.

7. Like others commenting on the original post, I say that the wording of “photocopying 169 sheets” makes this unsolvable. If you photocopy a sheet of paper, you could get anything from one to infinite copies. The wording used only tells you that the report itself uses 169 sheets and nothing about the number of copies.

If the wording was “returns with 169 sheets” then it can be solved as John took a 13 page report and made 12 additional copies (assuming he didn’t throw the original away). Of course that would make the solution be 10 members of the board.

1. Dave says:

I can’t believe how many people are being deliberately obtuse this week.

8. ‘a copy for all the board members’ implies that all the board members are going to have to share this copy. Total therefore is three: one for the board members to share, one for Linda, one for her mum.

1. simontaylor says:

But, 169 des not divide by three.

2. That would have made the puzzle unsolvable, though, as he didn’t ask how many copies were made — he asked how many board members there were. If you assume that the wording implies all board members will have to share a copy, the question becomes unanswerable. We can assume that a puzzle will be answerable and hold some clue to its solution, therefore we can dismiss the ‘one copy for all board members to share’ interpretation as invalid.

9. Mike Torr says:

Yep. Once again, a puzzle that’s clearly designed to provoke controversy in the comment with its ambiguities and subtle ‘deliberate-or-not?’ grammatical flaws.

I have long thought that this entire Friday Puzzle exercise is simply an extended psychology experiment, the point of which is to record our responses. Have you noticed that Richard never joins in with the comments? Of course he can’t: that would be interference with the outcome. I wonder when the results will be published…

1. David D says:

Its not an experiment. It just that in life there are loads of supercilious muppets out there!

2. Mike Torr says:

Of course I was being tongue-in-cheek about it being an experiment: my real point was that it is possible to word things far less ambiguously than Richard sometimes manages to do – and it seems to me that the majority of discussion in the comments for these puzzles centres around disagreements arising from this. If it is not an experiment, then I think he should take more care to write with precision; but perhaps my standards are too high. If I seem supercilious to you, it’s not something I put on deliberately and it’s not born of arrogance: it’s just the way I happen think about accuracy, that’s all.

10. The Masked Twit says:

This trivial question is so badly worded that only an idiot would waste time commenting on it………oops!

11. Darn – where did I come up with the idea that 169 was a prime number? That’s what I was working with all this time. Since it’s the square of a prime number, the problem IS simple.
And, btw, the CEO is not necessarily a member of the board. The CEO may be a “hired hand” that the board can fire, or give outragous bonuses to. (The boards should fire more of them.)

1. Anon says:

“One day she walked out of her modestly sized *boardroom*….”
Not that it matters as Linda asks for a copy for all board members “*plus* one copy for me…..”

12. Mickey D says:

Yes solved it in 2 secs!

13. Jo says:

Got it in a minute or so, but thought I must have missed something as I normally don’t work the Friday puzzles out in time!
For those wondering where the original goes, that one has to be a ‘clean’ copy for the files – the copies for the board members etc. are for them to scribble on and dissect if they so wish.

14. Chris says:

1. a “big company” can certainly have 169 board members. making that many copies in 11 minutes is unlikely but not impossible.
2. in common parlance, a set of copies is understood to include the original, so if the sec’y made 13 copies, there are now 14 copies to distribute – meaning 12 board members.

1. Dave says:

We were explicitly told that the boardroom was “modestly-sized”. Sometimes, you just have to read the question.

2. Anon says:

OK, Dave, so the boardroom is “modestly sized” what do we infer from that?
It could be modestly sized in absolute terms.
It could be modestly sized compared to the Colosseum or the egos of some of the contributors here.
It could be modestly sized for a company with a board of 169 members.
The company may have 169 members but only require a small number to be quorate – and generally only a few members attend.

If you read the question carefully you will notice that it is either not capable of solution or badly worded or indeed both.

[BTW I noticed 169 = 13×13 in about 20 seconds – so my grapes are sweet as ever.]

15. Anders says:

My answer: The CEO asked for COPIES for everyone, which means I left 13 as the original(not to be given to anyone), and then 2+10 copies, iow: 10 board members, unless the CEO was also a board member.

16. The correct answer could be any number. 169 was the number of pages photocopied, not the number of copied pages produced.

1. Anonymous says:

Fully agree. The wording states “photocopying 169 sheets” ie. 169 sheets were placed in the photocopier tray for copying.

For goodness sake Richard can you not take a little more care and not mislead your readers with such ambiguous wording.

What a complete waste of time

2. Julia says:

I usually get annoyed when people nitpick at the wording, but this time I found it so confusing that I couldn’t solve the puzzle.

3. It is not a “complete waste of time” for Richard; he is a psychologist, and he is fascinatedly analysing our responses.

17. One Eyed Jack says:

Let’s join the new millennium and e-mail them a pdf.

1. Please stop forcing me to use those P.(roprietary) D.(amned) F.(@&!&d-@%) files. I like the good old HTML or since you have to scan it, jpg or something. I do like your suggestion since the report was almost certainly done on a computer.
I read recently that the Government printed 4,500 copies of a 452 page document in which EVERY SINGLE WORD was crossed out – but that’s the rules – it was the old version of a bill!

18. Jeremy says:

I had the required answer, but for once I agree that the wording is poor.

The question says ‘make a copy of this report for all of the board members, plus one copy for me and one copy for my mother’.

That’s three copies, isn’t it? One for the board, one for Linda, one for Linda’s mother. It doesn’t say make the board members a copy each, it says make a copy for them all.

Then the secretary returns after ‘photocopying 169 sheets of paper’, presumably the length of the report.

(If I say I am going to photocopy something, then that something is the original, not the copy.)

So all we know is that the report is 169 pages long, that the secretary makes three extra copies from the original, and that making those copies (3 x 169 = 507 more pages) takes him 11 minutes.

I can’t see how I can work out how many board members there are from that.

19. Hugh Janus says:

If we had a couple of flagpoles, then the solution would be straightforward.

1. Chris says:

particularly since pi is exactly equal to 22/7.

2. No it isn’t. 22/7 is 3.142857 recurring, whereas pi is approximately 3.14159265358979323846.

20. Lazy T says:

I thought CEO stood for Children’s Entertainment Officer and the boardroom was where they played boardgames.
…still spotted the 13x13s tho’

21. Well, you’ve done a fine job of stating the problem better. It says “a copy” for the board, the 11 minutes is supposed to tell you something, it says he photocopied 169 sheets but doesn’t say how many times he photocopied each one, it doesn’t say if Linda is on the board or the board is merely for her company, is mom on the board, etc. I decided on 11 within seconds, but now I see how there can be 1,2,3,10,11,12,13, or almost any number

22. Emma says:

A disappointing answer. I realised very quickly that it has to be 13 * 13, but thought that was too straightforward and was expecting more of a clever twist. The wording is very ambiguous and I agree with those who say that ‘photocopying 169 sheets of paper’ would normally mean that there were 169 sheets in the original.

If we accept that he produced 169 photocopied sheets, then several other answers are possible if the original report was double-sided (26 ‘sides’ of writing on 13 sheets of paper) but some of the copies were single sided. For example, he could have produced 4 single-sided copies (26 sheets of paper each, 104 in total) plus 5 double-sided copies (13 sheets of paper each, 65 in total), giving a total of 9 copied reports implying 7 board members. (That’s not totally far fetched; it’s easy to mix up settings on copiers and produce double sided when you meant to produce single sided and vice versa.)

Going off on a diversion, I used to work at a charity for people with sight problems where some colleagues would need information in large print (differing font sizes, depending on the individual) so if something like that would increase the number of pages and throw off the equation completely.

23. Rudi says:

Sorry, you completely threw me by saying John “photocopied 169 sheets of paper”. I took this to mean the report must have been 169 pages long, since you don’t “photocopy” the copied pages, you only photocopy the originals. I took this as a self-evident fact, having had extensive experience of said office activity, and hence was totally stuck from there on. 😦

1. You are correct. It is for this reason that the puzzle is unanswerable.

24. dpettus says:

Simple solution. Ask the secretary. He obviousle already knows the answer. Why complicate things?

25. Will says:

I came up with either 11 or 167. Of course you did stipulate “Big Co.” so if it is 167 it must be loved by government.

26. I got the answer but only after laboring over the ambiguities of the wording of the puzzle. With due respect to Richard, dare I suggest that he carefully review his wording before publishing?

1. I think that the ambiguities in the wording of the puzzle were deliberate. How we interpret them is the whole point of why Richard asked us in the first place.

27. I’m going to gear this review to two types of people: current Zune owners that are considering an upgrade, and people trying to decide from a Zune plus an iPod. (There are other gamers worthwhile considering out there, just like the Sony Personal stereo X, but I hope this gives you enough details to make an educated decision of the Zune compared to gamers other than the iPod line as well.)

28. dharmaruci says:

we know 169 papers but not if some of them are doublesided copiers and some of them are two-up copiers for the more green board members.

therefore we do not know the number of the reports that where copiered. just we know the number of total papersheets. not even know the number of original sheets as some copiers may be double sided or reduced as i have already noted.

therefore this is not a clear puzzle to me.we can’not know the answer to any degree of satisfactory confidence without making unsatisfactory assumes.

29. surfnjer says:

The solution of 11 board members assumes “it is unlikely that there are over a hundred board members”.

30. Mint is done by Quicken. They closed down their online site after purchasing Mint.

31. I downloaded homebudget yesterday just starting to use it but it seems like what I\’ve been looking for