Sunday Times Teaser - 19 April

[Howard Somerset]

I've just had a quick look at today's teaser, and after only the first sentence, it seemed as if it's one just made for Dinos, having seen his conversations with Anne Robinson yesterday.

I've not even tried it yet, but especially for Dinos, and for anyone else interested who doesn't get the Sunday Times, here it is:

Between 200 and 300 men were asked to class themselves tall, dark or handsome, or any combination of these. A quarter felt they had none of these properties. The number including tall as a quality equalled the number including dark and the number including handsome. The number claiming to be tall, dark and handsome was half the number saying they were only dark and handsome. One more then this latter number said they were tall and dark; one more again claimed to be tall and handsome. This last group was exactly half the size of that saying they were only dark. How many claimed to be tall, dark and handsome?

[David Roe]

Solved. Answer by pm to Howard. (Clue - I'm definitely one of the tall, dark, handsome group. Of course. That's why I haven't put a photo - discourage fan mail. )

[Howard Somerset]

Solved. Answer by pm to Howard. (Clue - I'm definitely one of the tall, dark, handsome group. Of course. That's why I haven't put a photo - discourage fan mail. )

I won't read the PM yet, as I haven't tried the puzzle yet, but will do so in a few mins.

[Howard Somerset]

Tried it myself now, and I agree with the solution that David PMd to me.

[Phil Reynolds]

Hmm. I'm obviously crap at Venn diagrams, 'cos I tried this and found several valid solutions. For example: the number who are tall, dark and handsome could be 10, in which case 20 are dark and handsome, 21 are tall and dark and 22 are tall and handsome; 75 are tall (or dark, or handsome, since the puzzle tells us these numbers are the same). 54 are none of the three, and the total sample size is 216. However, the number who are all three could also be 11, 12, 13 or 14.

Obviously I've missed some vital constraint, so I'll be interested to hear the correct solution.

[Howard Somerset]

Hmm. I'm obviously crap at Venn diagrams, 'cos I tried this and found several valid solutions. For example: the number who are tall, dark and handsome could be 10, in which case 20 are dark and handsome, 21 are tall and dark and 22 are tall and handsome; 75 are tall (or dark, or handsome, since the puzzle tells us these numbers are the same). 54 are none of the three, and the total sample size is 216. However, the number who are all three could also be 11, 12, 13 or 14.

Obviously I've missed some vital constraint, so I'll be interested to hear the correct solution.

I think that what you've missed, Phil, is that every category must be an integer.

[Phil Reynolds]

Obviously I've missed some vital constraint, so I'll be interested to hear the correct solution.

I think that what you've missed, Phil, is that every category must be an integer.

Haha. Nope, no half-men in my solution.

[Howard Somerset]

Haha. Nope, no half-men in my solution.

How many categories have you got? Have you missed the one that most of the kids who come to me seem to miss when they first meet Venn diagrams?

[Phil Reynolds]

I've had another look and realised that first time around I'd cocked up due to missing the significance of the word "only" in "The number claiming to be tall, dark and handsome was half the number saying they were only dark and handsome". So I've redone it with that cock-up corrected, but I still can't get a unique solution. Having only recently sung the praises of Professor Ian Stewart (who taught me set theory), it's shaming to realise how little of either the technique or the notation I can now remember. Anyway, my Venn diagram looks like this:



T, D and H are the sets of those who responded they were tall, dark or handsome respectively. The answer we're trying to find, A, is the number who are all three (the intersection of the three sets). We're told that the cardinality (size) of T, D and H is the same. Using the information in the puzzle, we can fill in the sizes of the other segments of the diagram in terms of A. This results in a total size of 18A + 12 for the union of T, D and H.

Call the total sample size M; we're told that one quarter of this number did not belong to any of the three sets, so we know that 3M/4 = 18A + 12. Multiplying throughout by 4/3 gives M = 24A + 16. So the answer, A, is an integer for which the value of M satisfies the constraint 200 <= M <= 300. This works for A = 8, A = 9, A = 10 and A = 11.

Where am I going wrong?

[Kai Laddiman]

Yup Howard, he's done it...

[Phil Reynolds]

Yup Howard, he's done it...

If by that you mean "yes he's missed it", how about telling me what I've missed rather than just rolling your eyes?

[Dinos Sfyris]

It seemed as if it's one just made for Dinos, having seen his conversations with Anne Robinson yesterday.

Awesome. I clearly have fans in high places

pming you my solution now

[Howard Somerset]

... This works for A = 8, A = 9, A = 10 and A = 11.

Where am I going wrong?

Can't fault most of your work, Phil. In fact I got to the same stage. But as I said before, how many categories are there in the three ring venn diagram? You've only dealt with seven of them. There is an eighth. And only one of your values for A gives an integer value for the eighth category.

[Phil Reynolds]

But as I said before, how many categories are there in the three ring venn diagram? You've only dealt with seven of them. There is an eighth. And only one of your values for A gives an integer value for the eighth category.

Surely the eighth category is the set of respondents who are neither T, D nor H - i.e. the area in my diagram bounded by the rectangle but not within any of the circles? I didn't explicitly label this in the diagram with a value in terms of A, but as mentioned in my explanation it's M/4, i.e. 6A + 4 which is always an integer for integer values of A.

I'm clearly missing something trivially obvious and I'm bored now. As my reply to Kai indicated, I don't want hints - I just want to be told the answer!

[Howard Somerset]

ok Phil. I've looked a little closer at your formulae for the various categories. I think you may have misinterpreted the second of the following two sentences:

The number claiming to be tall, dark and handsome was half the number saying they were only dark and handsome. One more then this latter number said they were tall and dark; ...

The words lattter number refer to dark and handsome; I believe you've taken them to refer to tall, dark and handsome.

I did think when first reading the puzzle that this was one section that I'd have reworded had I been asked to proof read this puzzle before publication, but that's how it got printed, and so that's how I presented it here.

[Phil Reynolds]

ok Phil. I've looked a little closer at your formulae for the various categories. I think you may have misinterpreted the second of the following two sentences:

The number claiming to be tall, dark and handsome was half the number saying they were only dark and handsome. One more then this latter number said they were tall and dark; ...

The words lattter number refer to dark and handsome; I believe you've taken them to refer to tall, dark and handsome.

No, I interpreted "latter number" as the number who are only dark and handsome. The first sentence tells us this is 2A (so the number who are dark and handsome and possibly also tall is 3A). The latter number in the first sentence is 2A, and the second sentence tells us that 2A + 1 are tall and dark. The number who are only tall and dark is therefore that number minus the number who are all three, i.e (2A + 1) - A or A + 1.

[Dinos Sfyris]

Do you want me to pm you my workings and solution Phil?

[Phil Reynolds]

I did think when first reading the puzzle that this was one section that I'd have reworded had I been asked to proof read this puzzle before publication, but that's how it got printed, and so that's how I presented it here.

Ironically, that part of the wording seems to me to be perfectly clear. The bit I felt was possibly ambiguous was this:

The number including tall as a quality equalled the number including dark and the number including handsome

The "and" could be taken to mean either that the cardinality of T, D and H are all equal (which is how I've interpreted it), or that the cardinality of T equals that of the union of D and H.

[Phil Reynolds]

Do you want me to pm you my workings and solution Phil?

If that's quicker than Howard or Kai telling me what I've missed, yes please.

[Howard Somerset]

No, I interpreted "latter number" as the number who are only dark and handsome. The first sentence tells us this is 2A (so the number who are dark and handsome and possibly also tall is 3A). The latter number in the first sentence is 2A, and the second sentence tells us that 2A + 1 are tall and dark. The number who are only tall and dark is therefore that number minus the number who are all three, i.e (2A + 1) - A or A + 1.

ok Phil. I thought about tall and dark, and again this is a bit that I'd have reworded had I had any say in the puzzle's publication. tall and dark here does mean only tall and dark, and just a little later tall and handsome really does mean only tall and handsome. When doing the puzzle myself, I felt that was what was intended, and proceded accordingly. But I certainly feel there are grounds for it be interpreted in different ways. This is why proof reading for puzzle magazines or newspapers is not a trivial exercise, and one which I've always enjoyed doing. Pity I didn't get this one to do. Maybe in future I'll rewrite any puzzle I post here to turn it into the form I'd have liked.

[Howard Somerset]

The bit I felt was possibly ambiguous was this:

The number including tall as a quality equalled the number including dark and the number including handsome

The "and" could be taken to mean either that the cardinality of T, D and H are all equal (which is how I've interpreted it), or that the cardinality of T equals that of the union of D and H.

Agreed, Phil. That was the first the sections that I felt were not clearly worded.

[Kai Laddiman]

Anywayz, the answer's 3.5.

[Phil Reynolds]

tall and dark here does mean only tall and dark, and just a little later tall and handsome really does mean only tall and handsome.

Oh, ffs. Thanks, Howard. Dinos has now PMed me his solution (thanks Dinos) and that's also how he'd interpreted it. So neither my set theory nor my algebra was at fault - it was just a stupidly badly worded puzzle. I realise I appear to be in a minority here, but given three attributes X, Y and Z, the phrase "the number who are X and Y" in no way implies "...but not Z", which is what the question as set expects you to assume.

[Howard Somerset]

I agree Phil, it was certainly badly worded. And to repeat, I will consider rewording any puzzle I post in future.