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people have trouble with, until they accept it and move on, is that negative times negative is positive, not negative. Another, from a bit later in schooling, is that .999~ = 1, or that the square root of a negative number exists. That final one is really only justified by the fact that it works, which only experience can show, and that it forms an algebraic closure, which is an advanced concept. I myself didn't know whether zero was even or not until several years ago, because nobody really said whether it was at all clearly, and it wasn't obvious by itself. It wasn't until I saw it rigidly defined as "absolutely any integer multiple of 2, including 0" that I could be sure. Kids have seen enough patterns randomly broken in math by that time that it's hard to be certain unless someone says it clearly.
1184:"Bob met Alana on the street and forgot how old each of her kids are. She has 3 of them. Alana told Bob that the product of their ages is equal to 36. The sum of their ages is 1 higher than the house number of the white building across the street. Bob went home and couldn't figure it out and phoned Alana. Alana told him that she forgot to give him one more clue: the oldest one is a girl. Upon receiving that clue, Bob immediately figured out their ages."
1616:
a few weeks later to say he had failed, this behaviour repeated several times, and he eventually ended in an insane asylum (possibly in
Switzerland). I must admit that I wasn't really paying attention at the time, but if anyone could identify the chap I would be grateful - it's been nagging away at the back of my mind ever since!
1132:. But it doesn't remotely support your previous claim that "It's fairly arbitrary that we consider 0 automatically even". All it supports is the notion that the confusion is not, after all, too surprising a mistake, given the large number of peculiar-sounding things that students are asked to accept. --
1615:
Hi all, hope you can help. There was a BBC programme recently about a mathematician who spent his declining years (I think in the '30s of the last century) trying to solve some famous mathematical problem. He would write to his publisher announcing that he was close to a solution, only to write again
1159:
It has sometimes happened that I've been with mathematical nonexperts and have had to address the question of whether 0 is even or odd. They always seem more surprised by my assertion that there is absolutely no controversy about the answer than they are by the answer itself. So I conclude that for
892:
I am trying to find out if the number 0 (zero) is an even number or is it considered to be an odd number. My friends and I are in a major debate about this question. Thank you very much for your time and assistance in this matter. Please forward your answer directly to my email address: (email
1437:
Is Daniel's problem unclear, or is it my solution? 1-6-6 isn't wrong because it could be that one was born exactly 6 years ago, and one was born 6 years and 10 months ago. This one would be the eldest, though both their ages are 6. With my phrasing, there will be no single person of greatest age and
1705:
If it was Cantor, then someone has the story wrong, or at least is giving it the wrong nuances. Cantor was never insane, only depressed. I don't mean to minimize depression; I know it's a cruel and debilitating illness, but in my amateur estimation it's unlikely to lead to crankish behavior. Cantor
1057:
I think that's a stretch. The reason 0 is a special case in the unit/prime/composite thing is that it doesn't have a unique factorization, which is getting reasonably deep into the multiplicative structure of the naturals. For even/odd you don't need to do that; you just notice that the numbers are
1027:
Absolutely - in the more general context. How many people have heard of modular equivalence, though? For that matter, how many people are really all that familiar with the integers, other than those aspects related to basic arithmetic? Not so many, especially at the level I assume the poster is at.
1207:
The idea behind this riddle is that it is much easier for Bob than it is for us -- Bob knows what the number is, and we don't -- but we can work out the number, and their ages, based on Bob's reactions (whether or not he could figure it out at some point.) Try to work out what Bob is thinking, and
979:
Actually, one natural option would be (working in the nonnegative integers) to call positive integer multiples of 2 even, all other positive numbers odd, and 0 neither. It's the same way we deal with primes - primes 2 and up are primes, the rest are composites, 1 is a unit, and 0 is none of those.
1410:
Wow. I've known this riddle for a while and have never considered this point. This can be remedied by rephrasing it as "the one with the greatest age", though this sounds artificial. Of course, clarifying that ages are taken to be integers is necessary for the riddle to make sense, and is all the
943:
I confess I'm curious about how there could be much debate. Most people would agree that 3, 5, and 7 are odd, and that 2, 4, and 6 are even. Continuing the pattern of separation by two implies that 1 is odd and that 0 is even. A number is even if it is a multiple of 2, and odd if it is not; since
1098:
Another good example. I would take it as supporting my case, though, not refuting it. My point was that, from the perspective of someone who doesn't yet have a bird's-eye-view of the whole thing, it all seems a bit arbitrary, so why not one more to toss on the pile? Another example that a lot of
1479:
Actually, I have already mentioned that ages must be integers. But I think I disagree with your point. As a mathematical exercise, this problem is trivial. Its only virtue is the challenge of noticing the mathematical details hidden in the real-world explanations. One part of this is extracting
961:
I believe the logic is that the even numbers are multiples of 2, and the odd numbers are the others. In that case, if you figured a number could only be a multiple of 2 that was at least 2, it would be hard to categorize 0. It's not at all natural to consider 0 even, when working with positive
944:
2×0 = 0, zero must be even. Notice it does not matter what integer is doubled, whether even, odd, positive, negative, or zero; thus −6 is also even (2×(−3) = −6), and −7 is odd. (Often negative numbers are not discussed as odd or even, but there are times when they are important; an example is
1001:
Nonsense. It is anything but arbitrary that zero is even; it is essential! That's one reason I mentioned modular arithmetic. Throughout mathematics we depend on the rules for multiplying, adding, and subtracting evens and odds. If zero were not even, the consequences would be painful indeed.
1425:"...the oldest one is a girl" tells us that 1-6-6 is wrong and 2-2-9 is right (without that clue it could be either). I don't quite see how the problem proposed by Daniel would be remedied by using "...the one with the greatest age is a girl". Unless I'm completely missing something here.
1187:
Using some logic with that last hint, it seems to infer that two of the kids are twins, while the older one is not because the last clue seemed to make a huge difference in Bob solving the riddle. If that is the case, then the possible combinations are 1-1-36, 2-2-9 and 3-3-4.
1460:
If you really want to continue down this road, you'd want to say "the one with the greatest age, when expressed as a truncated integer," or some such. But come on, guys, it's a riddle, not an exercise in how finely you can slice a problem (with or without Occam's Razor).
242:. I can see that these are equal, but the explanation of the answer says that the first expression is differentiated to give the second using the chain rule. None of my math teachers have been able to figure out how this is done. Could someone give me some help? --
1480:"there is a unique greatest number" from "there is an oldest child" (which itself is extracted from "the oldest is a girl"). But in fact this deduction is not correct, so we are cheating our listeners if we use the original phrasing and expect them to make it. --
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If I understand this right, the solution hinges on the idea that two people can't be born in the same year and still have one that's older. Even if there's twins, there's an older one, and there doesn't have to be twins. —
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What I said was, considering only the counting numbers (1,2,3... with 0 tacked on as an afterthought) it makes as much sense to consider 0 a separate case, which is where the confusion seems to come from.
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Well, no we don't really "know that's impossible". We know that it's impossible to prove or refute from ZFC. That's not what Cantor was trying to do (in fact ZFC had not even been formulated). --
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According to the article, Cantor did in fact end up in a sanatorium. And he wasn't being a crank -- he truly thought he had the answer a few times... though we know that's impossible.
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Apologies, we answer questions right here, as noted at the top of the page. This gives everybody a chance to see the answer (and correct it if necessary). Also if you had looked up
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As I said, I didn't really pay much attention to the programme, so it is entirely possible that they said he died in a san, and I misrembered this as being an asylum.
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My personal theory as to why so many people are confused about this (even a biology PhD I won't name!) is that they're remembering being taught that 0 is neither
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Oh, I get it. I didn't mean the readers of the reference desk, but rather people (friends, etc.) to whom we (as in, anyone) might pose the riddle in person. --
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The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the
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1720:, on the other hand, did go a bit further round the bend, but again I don't know of any problem that he kept claiming to be on the verge of solving.
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So that means if I find the sums of all of the possible combinations of the 3 ages, then the answer is the one with the repeating sum?
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The best word I could think of for "the people to whom we are posing the riddle". I'd be glad to hear better suggestions. --
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Oh, then in that case it means that before the "oldest one is a daughter clue", there must have been multiple answers?
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Prove that the following proposition is a tautology using Quine's method and standard logical equivalences.
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I can allow that that may be correct, if you're offering it as an explanation of why many people get this
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485:{\displaystyle {\frac {d({\frac {dy}{dx}})}{dy}}={\frac {d({\frac {dy}{dx}})}{dx}}\times {\frac {dx}{dy}}}
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Call me crazy, but "readers" has always been standard in newspaper and magazine articles.
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Oh okay, I get it now. Sorry, Meni, I'm really having a slow day here. Btw: listeners?
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I can not figure out whether the clue about the white building is relevant.
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numbers. It's only when working with all integers that it seems obvious.
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Yes. He couldn't solve it after those two hints -- what does that mean?
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The following is a simplification of a math riddle I have came across:
923:, in which it is stated that zero is considered an even number. --
1727:. Who knows, maybe we just didn't understand him. (Certainly we
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235:{\displaystyle {\frac {d^{2}y}{dx^{2}}}\times {\frac {dx}{dy}}}
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Try it now. I'm not quite sure how to link to other websites.
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cross off possibilities as you go. That list you've got will
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It's fairly arbitrary that we consider 0 automatically even.
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Is there one definite answer or are there multiple answers?
161:{\displaystyle {\frac {d}{dy}}\left({\frac {dy}{dx}}\right)}
79:
Welcome to the
Knowledge Mathematics Reference Desk Archives
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at the bottom somewhere. Just Ctrf+F "insurance salesman".
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if you then substitute dy/dx in for f(x), then you'll have;
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nonmathematicians the answer is really not obvious! --
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a pattern, and it unambiguously assigns 0 to the evens.
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1520:I don't know if it's better but . . . Audience? :)
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1714:, so it's conceivable the story derives from that.
919:here on Knowledge, you'd have been shown along to
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691:{\displaystyle (P\to Q)\land (Q\to R)\to (P\to R)}
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832:a) What is Quine's method? Can't find much on it
1592:Aahhh... Indeed. <rubs chin thoughtfully: -->
1356:That link gives me a "page not found" message.
763:{\displaystyle (P\to Q)\land (Q\to R):(P\to R)}
1685:Yes that's it, here it is on the BBC website
1225:Is the clue about the house number relevant?
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1335:Sorry, Didn't realize it was a question.
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1411:more important with this phrasing. --
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1643:I'm HUGELY impressed! Thank you :)
835:b) Did I do anything right there
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1438:thus 1-6-6 is indeed wrong. --
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893:removed) thanks again.
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1731:didn't understand him.) --
877:then it is a tautology. --
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1374:Thanks, that works now.
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287:
282:
279:
276:
273:
270:
257:
228:
225:
220:
217:
211:
203:
199:
195:
190:
185:
181:
156:
150:
147:
142:
139:
133:
126:
123:
119:
105:
102:
100:
97:
95:
91:
90:
82:
81:
71:
70:
64:
48:
41:
40:
31:
15:
14:
13:
10:
9:
6:
4:
3:
2:
1793:
1778:
1775:
1771:
1770:
1769:
1766:
1762:
1761:
1760:
1759:
1752:
1749:
1745:
1744:
1743:
1742:
1741:
1740:
1737:
1734:
1730:
1726:
1723:Then there's
1722:
1719:
1716:
1713:
1709:
1704:
1703:
1694:
1691:
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1684:
1683:
1682:
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1625:
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1487:
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1397:
1396:
1393:
1390:Thanks guys.
1380:
1377:
1373:
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1371:
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1364:
1363:
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1343:
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1338:
1334:
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1328:
1317:
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1248:
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1222:
1221:
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1200:
1195:
1192:
1189:
1185:
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1157:
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1153:
1152:
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1138:
1135:
1131:
1127:
1126:
1125:
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1122:
1121:
1120:
1119:
1118:
1117:
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1105:
1102:
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1094:
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1019:
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1017:
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1000:
999:
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960:
959:
958:
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942:
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938:
935:
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918:
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913:
912:
910:
906:
902:
898:
887:
883:
880:
876:
872:
868:
864:
860:
856:
855:
854:
852:
848:
844:
840:
833:
830:
817:
814:
811:
805:
799:
787:
781:
770:
754:
748:
742:
736:
730:
724:
718:
712:
701:
698:
682:
676:
664:
658:
652:
646:
640:
629:
619:
616:
597:
594:
589:
586:
580:
572:
568:
564:
559:
554:
550:
543:
537:
534:
523:
520:
515:
512:
503:
493:
476:
473:
468:
465:
459:
453:
450:
439:
436:
431:
428:
419:
413:
407:
404:
393:
390:
385:
382:
373:
363:
360:
343:
340:
335:
332:
326:
320:
317:
309:
303:
300:
294:
288:
285:
277:
271:
268:
258:
255:
251:
250:
249:
248:
245:
226:
223:
218:
215:
209:
201:
197:
193:
188:
183:
179:
154:
148:
145:
140:
137:
131:
124:
121:
117:
103:
98:
96:
88:
84:
83:
80:
77:
76:
68:
61:
57:
53:
47:
42:
39:
35:
27:
23:
19:
1728:
1707:
1660:Could it be
1628:Georg Cantor
1626:Sounds like
1614:
1594:Black Carrot
1551:Black Carrot
1522:Zain Ebrahim
1499:Zain Ebrahim
1428:Zain Ebrahim
1389:
1367:Zain Ebrahim
1349:Zain Ebrahim
1337:Zain Ebrahim
1324:
1255:
1209:
1196:
1193:
1190:
1186:
1183:
1180:
1129:
1101:Black Carrot
1066:
1059:
1030:Black Carrot
982:Black Carrot
964:Black Carrot
901:4.239.234.26
891:
874:
870:
866:
862:
843:91.84.143.82
834:
831:
771:
702:
699:
630:
627:
244:superioridad
107:
94:
78:
1666:PrimeHunter
1177:Math Riddle
895:—Preceding
837:—Preceding
99:September 5
67:September 6
46:September 4
26:Mathematics
1748:DuncanHill
1718:Kurt Gödel
1690:DuncanHill
1645:DuncanHill
1618:DuncanHill
1392:Acceptable
1376:DuncanHill
1358:DuncanHill
1327:Acceptable
1296:Acceptable
1281:Acceptable
1266:Just one.
1258:Acceptable
1227:Acceptable
1212:help you.
1199:Acceptable
921:0 (number)
254:chain rule
1774:Trovatore
1765:Gscshoyru
1733:Trovatore
1632:Gscshoyru
1313:Gscshoyru
1268:Gscshoyru
1244:Gscshoyru
1214:Gscshoyru
1134:Trovatore
1071:Trovatore
857:Possibly
615:Richard B
56:September
50:<<
1729:at least
1464:LarryMac
1197:Thanks.
926:LarryMac
897:unsigned
839:unsigned
24: |
22:Archives
20: |
1630:to me.
1162:Dominus
89:pages.
1470:| Talk
1403:Daniel
1060:called
932:| Talk
879:CiaPan
1311:Yes.
1130:wrong
1004:KSmrq
950:KSmrq
948:.) --
69:: -->
63:: -->
62:: -->
44:<
16:<
1662:this
1568:talk
1512:talk
1486:talk
1444:talk
1417:talk
917:Zero
905:talk
875:true
869:and
847:talk
700:So,
252:The
168:for
1708:was
1210:not
871:not
867:and
60:Oct
52:Aug
1664:?
1570:)
1514:)
1488:)
1461:--
1446:)
1419:)
1002:--
907:)
865:,
863:or
849:)
803:→
794:→
785:→
752:→
734:→
725:∧
716:→
680:→
671:→
662:→
653:∧
644:→
613:.
581:×
460:×
327:×
210:×
58:|
54:|
1566:(
1510:(
1484:(
1442:(
1415:(
903:(
845:(
818:?
815:?
812::
809:)
806:R
800:P
797:(
791:)
788:R
782:P
779:(
758:)
755:R
749:P
746:(
743::
740:)
737:R
731:Q
728:(
722:)
719:Q
713:P
710:(
686:)
683:R
677:P
674:(
668:)
665:R
659:Q
656:(
650:)
647:Q
641:P
638:(
598:y
595:d
590:x
587:d
573:2
569:x
565:d
560:y
555:2
551:d
544:=
538:y
535:d
530:)
524:x
521:d
516:y
513:d
507:(
504:d
477:y
474:d
469:x
466:d
454:x
451:d
446:)
440:x
437:d
432:y
429:d
423:(
420:d
414:=
408:y
405:d
400:)
394:x
391:d
386:y
383:d
377:(
374:d
344:y
341:d
336:x
333:d
321:x
318:d
313:)
310:x
307:(
304:f
301:d
295:=
289:y
286:d
281:)
278:x
275:(
272:f
269:d
227:y
224:d
219:x
216:d
202:2
198:x
194:d
189:y
184:2
180:d
155:)
149:x
146:d
141:y
138:d
132:(
125:y
122:d
118:d
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