#27000  Tue May 07 2002 11:52 PM
The Sunday Times .. Brain Teaser X

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Loc: Sydney oz downunder

another one from the mid to late 70s quote: Prime Birthdays
At our local the other night I made the acquaintance of a chap called Michael and his wife Noelle. We talked a good bit about our respective families, our activities and our hobbies. When I happened to mention my interest in mathematical puzzles to Michael he said he knew one at first hand which might give me some amusement. He proceeded to tell me that although he himself, his parents (who were born in different years), Noelle and their children (featuring no twins, triplets, etc) had all been born in the nineteenhundreds and none of them in a leap year, the numerical relationship between their years of birth was nevertheless in two respects a bit unusual.
'In the first place,' he said, 'just one of us has our year of birth precisely equal to the mean of all the others' years of birth. But more remarkably the difference between my father's year of birth and that of any one of the rest of us is a prime, and the same is true of that of my mother. And she, like Noelle here, had no child after she passed her twenties.' And then with a grin he concluded, 'So now you'll know about the whole family and even, if you want, be able to figure out just how old Noelle is without having to be rude and ask her!'
In which year was Noelle born, and how many children does she have?

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#27001  Sun May 12 2002 12:52 PM
Re: The Sunday Times .. Brain Teaser X

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Registered: Mon Aug 14 2000
Posts: 616
Loc: Bridgnorth Shropshire England

gtho4,
Have spent a bit of time on this one and reckon to have found the main principle of the arithmetical problem.
Has the guy who originally set the problem been a touch unfair in using the word "children" when in fact Michael and Noelle have only one child????
Bannockburn

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#27002  Sun May 12 2002 08:15 PM
Re: The Sunday Times .. Brain Teaser X

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Registered: Mon Aug 14 2000
Posts: 616
Loc: Bridgnorth Shropshire England

gtho4,
Yes, there can be two children.
Bannockburn

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#27005  Mon May 13 2002 12:25 AM
Re: The Sunday Times .. Brain Teaser X

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Posts: 417
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Arggggg!! This question is driving me crazy! Just when I thought I'd finally found the answer, it was wrong again!! But I'm not giving up...yet.
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#27006  Mon May 13 2002 02:02 AM
Re: The Sunday Times .. Brain Teaser X

Mainstay
Registered: Mon Aug 14 2000
Posts: 616
Loc: Bridgnorth Shropshire England

gtho4 & Lo78,
I've got to the stage where I feel I know this question like the back of my hand.
Once before I suggested that there was either a mistake in a question or there was more than one solution only to have to bow my head in shame having not read a detail in the question correctly.
In this case I am feeling a bit more confident that there could be a mistake but am going to take a bit more time before I make a complete fool of myself.
In essence I have a solution that doesn't satisfy "the mean" criteria and another solution that satisfies "the mean" but not Noelle havng completed her childbirths at latest in her 20's.
The figures are very interesting to say the least and contain a couple of beautiful mathematical "funnies", if that is the way to put it, and are no doubt the "problems" that have to be seen to get anywhere near a solution.
My belief is that the figures are such that there are no alternative sets from which a solution can be derived, but two of the criteria of the question cannot be fulfilled at the same time, whilst all the others can.
As in the last instance I have asked a friend to have a look at my results and will get back to you.
Kindest Regards Bannockburn [ 05132002, 03:04 AM: Message edited by: Bannockburn ]

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#27007  Mon May 13 2002 10:14 AM
Re: The Sunday Times .. Brain Teaser X

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Hi Bannockburn and gtho4, I too feel like there might be a mistake in the question. It's either that, or I'm not up to solving it... quote: In essence I have a solution that doesn't satisfy "the mean" criteria and another solution that satisfies "the mean" but not Noelle havng completed her childbirths at latest in her 20's.
I know what you mean, Bannockburn! I found two solutions where all criteria are satisfied, except for the fact that the difference between the mother's year of birth and that of one of the children is not a prime...
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Errare humanum est, perseverarum diabolicum  Marcus Tulius Cicero

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#27008  Mon May 13 2002 12:49 PM
Re: The Sunday Times .. Brain Teaser X

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Registered: Mon Aug 14 2000
Posts: 616
Loc: Bridgnorth Shropshire England

gtho4 & Lo78,
I can get the whole thing to work out in as much that the difference in the "years of birth" are all primes, not forgetting that the difference in the "years of birth" between Michaels mother and father must, within the question, also be a prime.
I am sure that Michael and Noelle had but 2 children. 3 children would go OTT on the restriction of Noelle having completed childbirth within her 20's, again not forgetting that within the arithmetic the number 30 can be used, as this indicates Noelles 30th year and she can in fact be 29 years old.
Lo78  Without doubt you have picked up on the above already and also the facts that 2 is ( contrary to some beliefs ) a prime number, and also that 1900 is not a leap year
These are the "funnies" I referred to.
I think we are of like minds and please be assured that I am more than willing to give the details my analysis, for what is worth, but refrained from doing so in order that those who love the challenge could carry on for themselves.
I reckon there is a problem????????
Kindest Regards To You Both ( et al )
Bannockburn [ 05132002, 01:52 PM: Message edited by: Bannockburn ]

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#27009  Tue May 14 2002 12:27 AM
Re: The Sunday Times .. Brain Teaser X

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Bannockburn, I did assume that 2 is a prime as well, and that 1900 is not a leap year. I actually "overlooked" the fact that Noelle might be already in her 30th year... Oops! Don't think it would have helped me much, though. quote: I am sure that Michael and Noelle had but 2 children.
My English is leaving me in the lurch at this point, sorry! What does this sentence mean? I mean, I would have understood if it weren't for the quite essential word "but"...
If it's okay with ghto4, maybe we should just go through the whole solving process step by step here, to see if we can work it out together? Or we could PM?
Lo
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#27011  Tue May 14 2002 09:46 AM
Re: The Sunday Times .. Brain Teaser X

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Registered: Thu Sep 30 1999
Posts: 1006
Loc: Concord California USA

I've spent quite a lot of time working on this too, and like the others seem to be hitting my head against a brick wall. George, there are a couple of things that I'd like to establish to make it a little easier to arrive at a solution. First; because the question was originally printed in the mid to late nineteenseventies, can we assume that none of the birth years are after 1979, as would seem logical? Second; I'm a little fuzzy on this part of the question. "And she, like Noelle here, had no child after she passed her twenties". Does this mean that neither of the women gave birth to a child after she passed her twenties? Does this exclude the possibility of adoption of Michael and Noelle's children, or of Michael himself?
Now the math part. It would seem that the birth years of Michael's parents need to be separated by two years, as this is a prime and there are several sets of consecutive primes separated by two, such as 5 and 7, 11 and 13, 17 and 19, 29 and 31, 41 and 43, 59 and 61, and 71 and 73. Any other prime number separation in the birth years of Michael's parents seems impossible given the fact that each needs to be separated by a prime from all the others. Also, I think the oldest person needs to have been born between 1900 and 1906, otherwise the youngest person would be born born after 1979, which I don't think is possible, given the time the question was originally printed. Is any of this on the right track?

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#27012  Tue May 14 2002 11:03 PM
Re: The Sunday Times .. Brain Teaser X

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Tim, I'm glad to know Bannockburn and I are not the only ones having difficulties solving the question ! quote: It would seem that the birth years of Michael's parents need to be separated by two years
I came up with this too, seems like the only possibility to me.
quote: I think the oldest person needs to have been born between 1900 and 1906, otherwise the youngest person would be born born after 1979
Was this an assumption you made before starting your calculations, or afterwards? Because after reading the question it seems possible that both the mother and Noelle have had children in their very early twenties. Therefore the oldest person can be born after 1906. But if you came up with the quoted assumption after calculating, I'd like to know what you based it upon! Maybe it helps my understanding of the problem!
I know you were waiting for george to answer your questions on the original question itself, but I just wanted to give you my point of view.
Lo
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Errare humanum est, perseverarum diabolicum  Marcus Tulius Cicero

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#27013  Tue May 14 2002 05:33 PM
Re: The Sunday Times .. Brain Teaser X

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Registered: Mon Aug 14 2000
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Loc: Bridgnorth Shropshire England

gtho4 ( et al ), I think we are collectively agreed, in my case in took some time to deduce, that we are looking at a set of "years of birth" as follows. This set of numbers are, I am convinced, the only set that satisfy the requirement that the differences back to Michael's parents for all the others involved, are all prime numbers. Not only that it satisfies what I earlier called my "funnies" that 2 is a prime number and 1900 is not a leap year. So here is the list: 1900 1902 1919 1931 1943 1961 1973 It goes without saying that 1900 and 1902 must be "allocated" to Michael's mother and father, and again it goes without saying it doesn't really matter within the arithmetic as to which is the elder. I have not entirely preplanned this post, so for convention, should it be necessary, I will assume that Michael's father is the eldest, but in fact it does not really matter. The question could now well be asked "why start at the year 1900", and in asking this question I am sure we can all see from the above figures that it is necessray to remember that from any year "allocated" to Noelle it is impossible to jump 3 ahead and maintain the criteria that she had completed childbirth in her 20's ( ie 30 or less),and also the fact that we have been told by my good friend "gtho4" that the word "children" within the question is correct. Therefore Michael and Noelle can only have 2 children. There are therefore specifically a total of 6 persons involved. We have 7 years of birth to choose from, 2 of which are already allocated to Michael's mother and father, so we have to select 4 from the remaining 5, that will satisfy the parameters of the question, and I am going to venture to suggest that this cannot be done, however it is necessary, first of all, to prove that we are "locked in" to these years. To save "scrolling back" might I repeat the years. 1900 1902 1919 1931 1943 1961 1973 We have to choose 4 out of the lower 5. If the year 1900 is increased by 1 to 1901 then 1931 and 1943 become 1932 and 1944 respectively and any selection of 4 from 5 gives us a leap year' which is not allowed from within the question. I am sure you don't need me to explain the results of adding 2 or 3, and adding 4 makes Michael's father year of birth a leap year. Adding 5, 6, 7 & 8 naturally yield the same results. I therfore firmly believe that these "years of birth" are the only ones available to us from which to derive a solution. I have "gone on" long enough but would finally say this. Either Michael or Noelle must be the "mean", so they therefore from within the question, must be allocated different years of birth. As far as I can see there is no combination, that satisfies at the same time, the "mean" and "childbirth" requirements. gtho4 ( my friend )  If the answer you have is 2 children and that Noelle's year of birth is either 1931 or 1943 I think I can prove both wrong. We know each other well enough that this is only a bit of fun and no doubt you will make me hang my head in shame again. Kindest Regards To You All Bannockburn The Ignorant [ 05142002, 07:05 PM: Message edited by: Bannockburn ]

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#27014  Tue May 14 2002 05:47 PM
Re: The Sunday Times .. Brain Teaser X

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Registered: Mon Aug 14 2000
Posts: 616
Loc: Bridgnorth Shropshire England

gtho4,
I've just suddenly thought about something???
Has the chap who originally set this question in the "Sunday Times" been sufficiently devious to include another "funny".????
Michael and Noelle could have had 2 children within one twelve month period that were not twins???????
Should that be the case then the arithmetic will have to wait until tomorrow.
Bannockburn [ 05142002, 06:50 PM: Message edited by: Bannockburn ]

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#27016  Wed May 15 2002 02:04 AM
Re: The Sunday Times .. Brain Teaser X

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Registered: Tue Apr 16 2002
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Bannockburn, I'm impressed! The thought of the children having to be apart only nine months crossed my mind very very briefly when I was reading the question for the first time. But I never thought of it again ! I'm glad you did, though ! I don't have the opportunity to calculate anything today, so I'll just wait and see what you come up with... Lo
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#27017  Wed May 15 2002 03:56 AM
Re: The Sunday Times .. Brain Teaser X

Mainstay
Registered: Mon Aug 14 2000
Posts: 616
Loc: Bridgnorth Shropshire England

gtho4 ( et al ), I'll kill that man from The Sunday Times. A "double funny". Noelle actually had two children in each of two years giving a total of four and she herself was born in 1943??? Michael's Father 1900 Michael's Mother 1902 Michael 1931 Noelle 1943 1st Child 1961 2nd Child 1961 3rd Child 1973 4th Child 1973 Please don't tell me this is wrong??? Bannockburn

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#27018  Wed May 15 2002 04:08 AM
Re: The Sunday Times .. Brain Teaser X

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Of course gtho4 is the only one with the final answer, but it looks okay to me! Bravo ! This definitely was a tricky question!
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Errare humanum est, perseverarum diabolicum  Marcus Tulius Cicero

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#27020  Wed May 15 2002 01:12 PM
Re: The Sunday Times .. Brain Teaser X

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Registered: Thu Sep 30 1999
Posts: 1006
Loc: Concord California USA

Whew! Outstanding work Bannockburn! I must admit that it never occurred to me that Noelle could have had more than one child that weren't twins in a twelve month period. My hat is off to you! [ 05152002, 02:18 PM: Message edited by: tim10001 ]

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