Article 15918 of sci.math: From: wsadjw@rw7.urc.tue.nl (Jan Willem Nienhuys) Newsgroups: sci.math Subject: My little village's birthday problem Message-ID: <6016@tuegate.tue.nl> Date: 23 Oct 92 09:43:34 GMT Reply-To: wsadjw@urc.tue.nl Organization: Eindhoven University of Technology, The Netherlands Lines: 26 (A puzzle). When my son was born recently I went to the town hall of my little village to register him. When he was entered into the computer, the civil servant said: that's great! now our town can celebrate someone's birthday really every day of the year! I asked them: Really, also on leap days? Yes, she said, we've three people who have their birthday on a leap day. I asked her how many people our village had. She told me the number. On the way home I started thinking. What would be the chance that a random group of so many (I deducted the three leap day people) would have a birthday on every day of the year? In other words, which fraction of all maps [1..N] -> [1..365] are surjections? Fascinated I stopped my bike, sat down on the side of the road, whipped out my old calculator, and found that the fraction was as near to 0.5 as possible: any other N would have given a chance farther away from 0.5. When I came home, I redid the calculations without any approximations, and found that the actaul chance is just over 0.5 (but very very little). How many people live in my town (excluding the leap birthdays)? (Problem courtesy Aart Blokhuis. Afficionados know of course the answer to the ordinary birthday problem: at which number of people the chance that there will be two with coinciding birthdays exceeds 0.5? THAT answer is 23.) Article 15985 of sci.math: From: clong@remus.rutgers.edu (Chris Long) Newsgroups: sci.math,rec.puzzles Subject: Re: My little village's birthday problem (SPOILER) Message-ID: Date: 24 Oct 92 09:17:14 GMT References: <6016@tuegate.tue.nl> Followup-To: sci.math Organization: Rutgers Univ., New Brunswick, N.J. Lines: 18 In article <6016@tuegate.tue.nl>, Jan Willem Nienhuys writes: > How many people live in my town (excluding the leap birthdays)? Use inclusion-exclusion to get that for n people p(n) = \sum_{i=0}^n (-1)^i * (365 C i) * (1 - i/365)^n. We can approximate (1 - i/365)^n by e^{i*n/365} and so p ~ \sum_{i=0}^n (365 C i) * (-e^{n/365})^i = (1 - e^{n/365})^n. An easy calculation gives that p(n) first exceeds 1/2 when n=2288, and this should be quite accurate, but I'll check when I have the time. Let me tell you about the town I'm from. In this town no worker has any weekends or holidays off, except if another worker has a birthday, then everyone has off. Luckily for productivity, the number of workers is such that the expected number of worker-days is maximized. How many people work in my town, ignoring leap days? -- Chris Long, 265 Old York Rd., Bridgewater, NJ 08807-2618 Article 15996 of sci.math: Newsgroups: sci.math,rec.puzzles From: jbaez@riesz.mit.edu (John C. Baez) Subject: Another sort of birthday puzzle Message-ID: <1992Oct24.183017.13325@galois.mit.edu> Nntp-Posting-Host: riesz Organization: MIT Department of Mathematics, Cambridge, MA References: <6016@tuegate.tue.nl> Date: Sat, 24 Oct 92 18:30:17 GMT Lines: 10 Here's a birthday puzzle I recently heard on PBS. I hope the rec.puzzle crowd forgives me if this old hat to them. (I just recently tuned into rec.puzzles and was amused to find that some of my information on the Voynich Ms. had found its way into the FAQ there!) Two twins (the usual number) celebrated their birthdays one year, and it was rather unusual in that one celebrated his birthday two days before the other! What's more, the younger one celebrated his birthday first! How did this happen? Article 16003 of sci.math: Newsgroups: sci.math,rec.puzzles From: russotto@eng.umd.edu (Matthew T. Russotto) Subject: Re: Another sort of birthday puzzle Message-ID: <1992Oct24.211256.27129@eng.umd.edu> Date: Sat, 24 Oct 92 21:12:56 GMT Organization: College of Engineering, University of Maryland, College Park References: <6016@tuegate.tue.nl> <1992Oct24.183017.13325@galois.mit.edu> Lines: 21 In article <1992Oct24.183017.13325@galois.mit.edu> jbaez@riesz.mit.edu (John C. Baez) writes: >Two twins (the usual number) celebrated their birthdays one year, and it >was rather unusual in that one celebrated his birthday two days before >the other! What's more, the younger one celebrated his birthday first! >How did this happen? They crossed the International Date line and the 12:00 midnight line flying west between the birth of the two twins. i.e. twin 1 was born at 12:01 on Friday. They crossed the midnight line, which made it 11:59 on Thursday. They then crossed the date line, making it Wednesday. Then the second twin was born, some minutes later according to duration, but one day and several minutes earlier according to clock time. -- Matthew T. Russotto russotto@eng.umd.edu russotto@wam.umd.edu Some news readers expect "Disclaimer:" here. Just say NO to police searches and seizures. Make them use force. (not responsible for bodily harm resulting from following above advice) Article 16024 of sci.math: From: rmjarvis@cco.caltech.edu (Robert Michael Jarvis) Newsgroups: sci.math,rec.puzzles Subject: Re: Another sort of birthday puzzle Date: 25 Oct 1992 09:35:47 GMT Organization: California Institute of Technology, Pasadena Lines: 17 Message-ID: <1cdppjINN4bf@gap.caltech.edu> References: <1992Oct24.183017.13325@galois.mit.edu> <1992Oct24.211256.27129@eng.umd.edu> In article <1992Oct24.211256.27129@eng.umd.edu> russotto@eng.umd.edu (Matthew T. Russotto) writes: >They crossed the International Date line and the 12:00 midnight line >flying west between the birth of the two twins. i.e. twin 1 was >born at 12:01 on Friday. They crossed the midnight line, which made >it 11:59 on Thursday. They then crossed the date line, making it >Wednesday. Then the second twin was born, some minutes later >according to duration, but one day and several minutes earlier >according to clock time. Actually, it is possible to get another day's difference in there. If the first twin was born at 12:01 Friday, March 1, and the second twin was born at 11:59 on Thursday, February 28, then every four years (except for the obvious exceptions) they would celebrate their birthdays 3 days apart, since there would be an extra day in between. Namely February 29. Mike. Article 16023 of sci.math: Newsgroups: sci.math,rec.puzzles From: msb@sq.sq.com (Mark Brader) Subject: Re: Another sort of birthday puzzle Message-ID: <1992Oct25.080339.16732@sq.sq.com> Organization: SoftQuad Inc., Toronto, Canada References: <6016@tuegate.tue.nl> <1992Oct24.183017.13325@galois.mit.edu> Date: Sun, 25 Oct 92 08:03:39 GMT Lines: 34 > Here's a birthday puzzle I recently heard on PBS. I hope the rec.puzzle > crowd forgives me if this old hat to them. ... > > Two twins (the usual number) celebrated their birthdays one year, and it > was rather unusual in that one celebrated his birthday two days before > the other! What's more, the younger one celebrated his birthday first! > How did this happen? Well, this isn't old hat on rec.puzzles, but Games magazine subscribers may recognize it. From the new (December 1992) issue: # A man spends several days reading about midget apartment dwellers # who can't reach their elevator buttons; blind tightrope walkers # murdered by bandleaders stopping the music too soon; goldfish with # human names lying dead in pools of water; and lighthouse keepers # who forget to turn on their beacons and commit suicide when their # negligence causes ships to crash. How come? # # The answer is simple: The man was judging a "How Come?" contest and # reading classic puzzles submitted by readers. The birthday puzzle cited above was the winning entry in the contest (the wording in the magazine is slightly different, but the content is identical); it was submitted by Judy Dean of Koror, Palau. One posting with an attempted solution has reached here so far; it's wrong. There isn't a "midnight line" that you can cross so that the time changes from 12:01 to 11:59. -- Mark Brader "One of the lessons of history is that nothing SoftQuad Inc., Toronto is often a good thing to do and always a clever utzoo!sq!msb, msb@sq.com thing to say." -- Will Durant This article is in the public domain. Article 16097 of sci.math: From: dseal@armltd.co.uk (David Seal) Newsgroups: sci.math,rec.puzzles Subject: Re: Another sort of birthday puzzle Message-ID: <8831@armltd.uucp> Date: 26 Oct 92 12:10:13 GMT References: <1992Oct25.080339.16732@sq.sq.com> Distribution: sci Organization: A.R.M. Ltd, Swaffham Bulbeck, Cambs, UK Lines: 47 In article <1992Oct25.080339.16732@sq.sq.com> msb@sq.sq.com (Mark Brader) writes: >> Here's a birthday puzzle I recently heard on PBS. I hope the rec.puzzle >> crowd forgives me if this old hat to them. ... >> >> Two twins (the usual number) celebrated their birthdays one year, and it >> was rather unusual in that one celebrated his birthday two days before >> the other! What's more, the younger one celebrated his birthday first! >> How did this happen? > >One posting with an attempted solution has reached here so far; it's >wrong. There isn't a "midnight line" that you can cross so that the >time changes from 12:01 to 11:59. That isn't the problem - there is such a line. It's not a fixed line - it's moving west at the correct rate to go around the world once every 24 hours. If you go west faster than that, your local time will run backwards, and as you cross the "midnight line", your local time will go backwards through midnight: it is perfectly possible for it to go from 12:01 a.m. on one day to 11:59 p.m. on the previous day. (If you want to use time zones rather than simple local time, then (ignoring some of the odder time zones around) the midnight line remains stationary at each time zone boundary for an hour, then instantaneously leaps to the next time zone boundary to the west. You can still travel westwards through the midnight line and have the time go back - it's just that it now has to go back by a full hour, not a couple of minutes.) The real problem with the proposed solution is that it tries to make use of both the midnight line and the International Date Line. This doesn't work because the two lines work in opposite directions: going westwards through the midnight line moves the date back, while going westwards through the International Date Line moves the date forward. This is the whole purpose of the International Date Line: it compensates for the date difference caused by going through the midnight line, allowing dates to be defined consistently over the whole globe. (Consistent in the sense that if you circumnavigate the world, you find that the date you think it is and the date the locals think it is agree at the end of the journey: if your date changed the same way at the midnight line and at the International Date Line, a circumnavigation would put you two days out of synch with the locals.) David Seal dseal@armltd.co.uk All opinions are mine only... Article 16053 of sci.math: From: sklar@picasso.ocis.temple.edu (Dave Sklar) Newsgroups: sci.math,rec.puzzles Subject: Re: Another sort of birthday puzzle Message-ID: <1992Oct26.011803.134@cronkite.ocis.temple.edu> Date: 26 Oct 92 01:18:03 GMT References: <1992Oct24.183017.13325@galois.mit.edu> Sender: news@cronkite.ocis.temple.edu (NetWork News (readnews)) Organization: Temple University Lines: 5 This is in this month's Games. The twins were born on a plane flying over the international date line. From feb.28-Mar.1. The particular year in the problem is a leap year. Dave Article 16079 of sci.math: From: M.Willis@ee.surrey.ac.uk (Mike Willis) Newsgroups: sci.math,rec.puzzles Subject: Re: Another sort of birthday puzzle Message-ID: <1992Oct26.144839.12256@EE.Surrey.Ac.UK> Date: 26 Oct 92 14:48:39 GMT References: <6016@tuegate.tue.nl> <1992Oct24.183017.13325@galois.mit.edu> Organization: University of Surrey, Guildford, Surrey, UK. GU2 5XH Lines: 24 In article <1992Oct24.183017.13325@galois.mit.edu>, jbaez@riesz.mit.edu (John C. Baez) writes: |> Here's a birthday puzzle I recently heard on PBS. I hope the rec.puzzle |> crowd forgives me if this old hat to them. (I just recently tuned into |> rec.puzzles and was amused to find that some of my information on the |> Voynich Ms. had found its way into the FAQ there!) |> |> Two twins (the usual number) celebrated their birthdays one year, and it |> was rather unusual in that one celebrated his birthday two days before |> the other! What's more, the younger one celebrated his birthday first! |> How did this happen? Well, an alternative solution. If you believe that children mature quicker in the woumb than once born, the second twin to be born is the elder. Now if they were born around midnight on 28th Feb, and it was not a leap year, then the younger child would have a birthday on the 28th Feb, the elder on 1st March. In a leap year there are two days between birthdays on 28th Feb and 1st March. You can also fiddle with crossing the date line, Midgits and Goldfish etc etc if you are so inclined. Mike Article 16126 of sci.math: From: achut@nairobi.Eng.Sun.COM (Achut Reddy) Newsgroups: sci.math,rec.puzzles Subject: Re: Another sort of birthday puzzle Date: 27 Oct 92 07:15:16 GMT Organization: Sun Microsystems Inc., Mountain View, CA Lines: 12 Message-ID: References: <6016@tuegate.tue.nl> <1992Oct24.183017.13325@galois.mit.edu> jbaez@riesz.mit.edu (John C. Baez) writes: >Two twins (the usual number) celebrated their birthdays one year, and it >was rather unusual in that one celebrated his birthday two days before >the other! What's more, the younger one celebrated his birthday first! >How did this happen? Alternate solution besides the int'l dateline & leap year trickery. The two were "twins", but not to each other! Each is member of a different pair of twins. -achut Article 16147 of sci.math: From: msb@sq.sq.com (Mark Brader) Newsgroups: sci.math,rec.puzzles Subject: Re: Another sort of birthday puzzle Message-ID: <1992Oct27.192417.27502@sq.sq.com> Date: 27 Oct 92 19:24:17 GMT References: <1992Oct24.183017.13325@galois.mit.edu> Organization: SoftQuad Inc., Toronto, Canada Lines: 19 > Alternate solution besides the int'l dateline & leap year trickery. > The two were "twins", but not to each other! > Each is member of a different pair of twins. That's very nice! The original version in Games magazine was worded so as to rule it out, but not the version posted here. I have another solution, but it's rather recherche. The twins were born, let us say, in England on January 15, 1800. (The exact details aren't critical.) As an adult, the older twin moved to Russia, where they used the Julian calendar until 1918, and began celebrating his birthday by that calendar, i.e. as January 4. From 1901 on, this would correspond to January 17 in the Gregorian calendar used in England. -- Mark Brader "Sixty years old and still pulling a train! SoftQuad Inc., Toronto That's more than I can say about most utzoo!sq!msb, msb@sq.com people I know." -- Frimbo This article is in the public domain. Article 16065 of sci.math: From: wsadjw@rw7.urc.tue.nl (Jan Willem Nienhuys) Newsgroups: sci.math Subject: Re: My little village's birthday problem (SPOILER) Message-ID: <6034@tuegate.tue.nl> Date: 26 Oct 92 10:00:36 GMT References: <6016@tuegate.tue.nl> Reply-To: wsadjw@urc.tue.nl Organization: Eindhoven University of Technology, The Netherlands Lines: 25 In article clong@remus.rutgers.edu (Chris Long) writes: #In article <6016@tuegate.tue.nl>, Jan Willem Nienhuys writes: # #> How many people live in my town (excluding the leap birthdays)? # #Use inclusion-exclusion to get that for n people p(n) = \sum_{i=0}^n #(-1)^i * (365 C i) * (1 - i/365)^n. We can approximate (1 - i/365)^n #by e^{i*n/365} and so p ~ \sum_{i=0}^n (365 C i) * (-e^{n/365})^i = #(1 - e^{n/365})^n. An easy calculation gives that p(n) first exceeds #1/2 when n=2288, and this should be quite accurate, but I'll check #when I have the time. Close, but wrong. # #Let me tell you about the town I'm from. In this town no worker #has any weekends or holidays off, except if another worker has #a birthday, then everyone has off. Luckily for productivity, the #number of workers is such that the expected number of worker-days #is maximized. How many people work in my town, ignoring leap days? It's the global village. The year is 1984. Big brother has decreed that all birthdays fall on January 1. :-( JWN Article 16069 of sci.math: From: clong@remus.rutgers.edu (Chris Long) Newsgroups: sci.math Subject: Re: My little village's birthday problem (SPOILER) Message-ID: Date: 26 Oct 92 13:33:13 GMT References: <6016@tuegate.tue.nl> <6034@tuegate.tue.nl> Organization: Rutgers Univ., New Brunswick, N.J. Lines: 13 In article <6034@tuegate.tue.nl>, Jan Willem Nienhuys writes: > In article Chris Long writes: > #(1 - e^{n/365})^n. An easy calculation gives that p(n) first exceeds > #1/2 when n=2288, and this should be quite accurate, but I'll check > #when I have the time. > Close, but wrong. A little program gives that n=2287, where p=.50037. As I said, the 2288 figure was an approximation, and a good one at that. -- Chris Long, 265 Old York Rd., Bridgewater, NJ 08807-2618