Tuesday, May 17, 2011

The cake paradox

There will be cake today
At the institute we have a (recent) tradition to bring home-made cake for the group. Each co-worker is assigned one week within he or she can freely choose one workday to bring the cake. So the actual day when there is cake will be a surprise to the others.
Unless... there is one problem when the process is viewed from a logical perspective.
Consider me having made a cake and planning to bring it in on Friday. Friday is the last workday in the week, so the others could predict the cake to be brought on Friday as soon as they see on Thursday that there is no cake. So I will not choose Friday, because it won't be a surprise on that day.
However, assuming totally logical actors, also Thursday is not an option, since the others will come to the same conclusion that Friday is off the list and they would know on Wednesday evening, when no cake appeared so far, that I will bring it on Thursday. So Thursday is not a surprise day either. We can iterate this argument and end up with the interesting situtation that I have to bring the cake on Monday, since all other days would not be a surprise. Having decided that even Monday is no surprise either.
So it looks like that it is impossible to bring a cake on a weekday as a surprise if everybody knows that I have to bring a cake within this week.
The paradox is interesting but it is also obvious that there is a difference between totally logical and natural actors. Asking people, they usually agree on the argument that Friday would be no surprise, but every other day would be. What do you think?

If you like this kind of puzzles, I recommend the book of Zbigniew Michalewicz and his son on Puzzle-Based Learning.


  1. I think there is a difference between a "logical" actor and a "rational" actor. A logical actor will be trapped by the paradox. A rational actor will assign a probability based on some scheme: for example an evenly-distributed probability for each remaining day (i.e. making Friday no surprise), historical patterns of cake-bringing, etc. In this situation, we naturally act like rational actors rather than logical actors.

  2. I think your proposed conclusion is not right. It is not impossible to bring a cake on a weekday as a surprise. Rather, from your argument just follows that it is not possible to surprise everybody every week. As mentioned above, using a probability distribution over the weekdays (and making public that you use such a distribution) will ensure a surprise in all weeks in which Friday is not the outcome. Assuming you work there infinitely long, you will surprise your friends infinitely often... that should be good enough.

    But if you really want to surprise them, try bringing cake twice a week ;-)