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Paradox Day - July 19, 2010
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A big thanks to Jason Dyer (The Number Warrior) for allowing me to use the 'Paradox Day' idea!

Somebody says that next year we shall have a paradox day and ... POOF!

So the cake is a lie and not a lie? But the cake is a lie! I saw it written! =)

It's not lie!

Best of all, you can have your cake AND eat it, too.

March 5th:
someone says 'what did you do last night?'
you say 'nothing'

Oh, goody: We can have our cake and eat it too! Or is that not have our cake and eat it, or have it and not eat, or.... Maybe all four possibilities on paradox day. With the help of an unobserved universe.

I didn't know Schrödinger could bake!

Of course, the logical gap behind this paradox has been uncovered: on December 30 you have additional information that you didn't have on December 29, namely that December 30 is not Paradox Day. And so on. So, on December 1, say, the information is not enough to decide whether December 2 will, or will not be, Paradox Day. Unless, of course, Paradox Day has already happened, on July 17. Unless, just to make things more interesting, you have two Paradox Days that year. Which would be appropriate.

Yay! someone understands why this is not a paradox! it's so hard to explain this to people...
I tried to do so once, but failed miserably...

Ok. I'll bite. What's wrong with the following.

Theorem 1. If the date of the paradox day is limited to dates in a set S, then the paradox day cannot occur on the last date in S.

Proof. If paradox day would occur on the last date of the set S, then it would not come unexpectedly. QED

First apply the theorem to the set S of all the dates in a calendar year to eliminate Dec 31st from the set. Then apply it to S = all dates save Dec 31st to eliminate Dec 30th ...

I do like Martin Gardner's (RIP) version of a village of 40 adulterers better.

You can't just eliminate it from the set.
The problem is, on December 29th, you don't know whether or not you will be celebrating on December 30th, and so the argument that excludes the 31st no longer applies. (Since this argument requires you to know that there is only one day remaining.)

It's only a paradox if the set only contains 2 days, in which case it's a trivial case of having two contradicting statements, and isn't really all that interesting...

But if you weren't celebrating on the 30th, you would know you were celebrating on the 31st, eliminating the surprise. Why is this not true?

As I said, it's only a paradox if there are two days left. But the inductive argument that seems to create a paradox in he other situations isn't valid.

But it is possible to have only one day left, no? Therefore the paradox could exist. Currently I am leaning toward the explanation presented by Jyrki after this one.

You say "this argument requires you to know that there is only one day remaining". I say "this argument requires you to know that there is only one possible date for the paradox day to occur remaining". It sounds to me that you are saying that we both can and cannot eliminate Dec31st? When asked for elaboration, you reply by repeating that "this step is invalid", when others have tried to explain, why it is valid :-)

I know, I know, it is difficult to come up with a counterexample to a step in an invalid proof of a true theorem, but we are not making progress...

My explanation to the apparent paradox is that the inductive argument uses a somewhat hazy meaning of the word "unexpected". I'm not at all positive that such a meaning could be defined using accepted formal logical systems. In other words, the concept of the paradox day is not well-defined (which is to some extent the whole point). Pointer to a textbook? Anyone?

The logic is similar to the famous

Theorem 2. There are no natural numbers without interesting properties.

Proof. Being "the smallest uninteresting natural number" is an interesting property. Therefore the set of uninteresting natural numbers mus be empty, for otherwise (by Peano axioms) it would have a smallest element.

That depends on whether 'the nth number which was the smallest uninteresting number until it was promoted to being interesting, and then a new number took over the title of smallest interesting number' is sufficiently interesting. Like paradox day, the concept of an interesting number is not well-defined. But you knew that. :)

Making one more attempt to understand why you think differently: Both you and bmonk refer to deductions done at specific dates, and seem to think that more information is accumulated during the year. The point of the inductive argument is that Dec 31st is eliminated well before the ides of March (of the previous year, if you like) simply because the paradox day would not surprise anyone on Dec 31st. For this reason we can rerun the argument early as many times as we wish. Eliminations are done well in advance. By logic alone - not by observations during the year,

Holy Crap! The paradox day ended up happening on my birthday. March 4th. That's a surprise!

Why not just say that paradox day has a 1/~365.25 chance of happening any given day? Then there could be more than one per year or one or zero. Don't tie it to once per calendar year and it can happen whenever, even dec 31 with equal surprise.

I think I understand why everyday is not paradox day now.(At least I think I understand what bmonk is trying to say)
but, I don't understand why March 4th is paradox day, by the way.

right, the only source we have stating that the cake is a lie is some crazy guy who also believes that the weighted companion cube can talk, besides you can clearly see the cake during the end credits

maybe if we re-evaluate the meaning of surprise, such that we don't know at the beginning of the year when paradox day will be. So technically if we get to December the 30th and we have not had paradox day then it sure is a surprise that paradox day is on the 31st. No one was expecting that 'cos it couldn't happen', therefore its a surprise.

What if we allow Paradox Day to not happen at all?
Then you can't be certain on Dec 30 that it will happen on Dec 31.
Of course, if we do that, it wouldn't be Paradox Day.

Great comic!

The problem with your hypothesis is simply this: there is already a Paradox Day during the calendar year. In 1982, Games Magazine ran a contest to come up with a valid holiday for the month of August, which has no holiday. The winning entry was Paradox Day. Paradox Day is observed on August 1 to celebrate the fact that August has no holidays. Of course, if Paradox Day is observed, then it cannot be. But if it is not observed, then it must.

But August 1 is the Swiss national day. :/

There are infinite ways to make a paradox day work, examples:
Having a paradox day outside of the calendar, like April 29th. Having the number of paradox days per year relation be different to 1.

the point about lack of information can also be resolved: lets say 365(or 366) people start tracking the paradox day starting at different days of the year, and exchange information with each other on when in the year it happened. So for each of the above individuals the paradox day has to happen on at least one of the days in this 365(or 366) day period, but can't happen on the last day of this period for each individual, therefore it cannot happen on any day of the year. nothing new in this but just refutes the lack of information argument :)

"for each of the above individuals the paradox day has to happen on at least one of the days in this 365(or 366) day period" That's not true, unless their 365-day period begins on January 1. Paradox Day doesn't necessarily happen on the same day each year, so it's perfectly possible that somebody who started tracking after Paradox Day of that year would not see a Paradox Day until after more than a year had elapsed. Paradox Days (if they could happen at all, which depends on the definition) could be up to 730 days apart. (If one is on Jan 1 of a leap year and the next is December 31 of the following year.)

We could also say that *every day* is paradox day. :-D

Today is Sweetmorn, the 62nd day of Bureaucracy in the Year of Our Lady of Discord 3176, btw.

March 4? Hot dog, that's my birthday!

No, the best of all, you both CAN'T have cake AND eat it too.

But who will decide which day Paradox Day is on any given year?

This is NOT a paradox. Hasnohat is correct. This is not another instance of the Adulterer problem because unlike the Adulterer problem, your information set at time t does not grow.

Adulterer Problem: There is a Village of 40 monogamous heterosexual couples. All of the men in the village are law abiding and perfect logicians. Everyone in the village cheats on their partner, and every man knows about the infidelity of every person except his own wife. By village law, if a man knew their wife was unfaithful, he should kill her that night. One day a grand priest who only speaks the truth declares that he knows there is at least 1 unfaithful wife. On the 40th night, every man kills his wife. Why?

Answer: Assume there is only 1 unfaithful wife. A man would always know of any woman's infidelity as long as that woman is not his wife. If that is true, there is a man who would not know ANY unfaithful women. Being a perfect logician, he would reason that HIS wife is the unfaithful one, and thus kill her that night. One night passes and no one was killed... If there was only 1 unfaithful wife, she would have been killed that night, so now the men assume there are 2 unfaithful wives. [THIS IS WHAT IS WRONG WITH THE PARADOX DAY QUESTION, SINCE WE CANNOT SEE INTO THE FUTURE, WE HAVE NO ADDITIONAL INFORMATION AS TIME PROGRESSES]. If this is true, there would be 2 men who know ONLY 1 unfaithful woman. Those men would kill their wives on the 2nd night. Continuing in this fashion, 39 nights pass when no one was killed... On the 40th day, all the men must assume there are 40 unfaithful wives. As there are only 40 women in the village, every man kills his wife that night.

Paradox Day: It is true that on Dec 30th, Dec 31th cannot be a paradox day. But on Dec 29th, you NO LONGER KNOW that Dec 31th cannot be a paradox day because the Dec 30th has yet to arrive with no Paradox day happening.

I hope this clarifies the situation. Feel free to email me if you have any issues with my argument.

I must disagree, this is a paradox if you are using proper logic.

The main premise is that the day cannot be known ahead of time and, as we all agree, once you arrive at Dec 30, we would know the holiday must be Dec 31 thereby eliminating this day as a possibility...permanently. We all know logically that it can't be the last day of the year, ever, from now until the end of time, so it doesn't require seeing into the future to know this. The only way that would be true is if we didn't know how long the year was and we awoke every day of December surprised that it's still not January. What I think is being missed is that we know ALL YEAR LONG that the holiday cannot be on Dec 31 as that would violate the very nature of the holiday, thus I cross it off my calender.

Therefore, when I get to Dec 29 I have the pre-knowledge that Dec 31 was never a possibility so Dec 30 is now the only option. But I can logically figure that out ahead of time too, as I just did in real life, so Dec 30 gets crossed off the calendar too. Rinse and repeat and you've just crossed off every day just as the cartoon suggests.

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