Hansel and Gretl have discovered a gingerbread cottage and are wondering whether to eat some of the tiles on its walls. A witch appears and tells them how they must go about it. “Each of you is to name a whole number between 0 and 100. Hansel’s must be odd and Gretl’s even. No conferring. Whoever chooses the lower number can eat twice that number of gingerbread tiles. Whoever chooses the higher number can eat the lower number.” So, for example, if Hansel chooses 57 and Gretl chooses 30, Hansel will get 30 tiles and Gretl will get 60.

This sounds fine, but the children have just had lessons in game theory and regard this as a non-cooperative game between rational utility maximizers. Gretl knows that Hansel will not choose 99, because 97 would leave him better off if she chose 98 and no worse off if she chose any other number. By the same reasoning, she will avoid 98 and choose 96. In her mind she can follow this train all the way to its end: Rationally, it seems, she must choose 2. Hansel, following it also, finds himself indifferent between 3 and 1. In the end he will receive a paltry two tiles and Gretl either one or four.

Is all of this sound? Gretl says, “There is something radically peculiar about trains of thought which proceed in the subjunctive. You are to work out what you would be rational to do, if I were to choose a number which I shall not choose. I am to do likewise, with each train of thought reproduced inside the other. What happens if either player derails a train by choosing in defiance of it? In that case it becomes radically unclear whether either player still has a rational choice.”

(Martin Hollis, “The Gingerbread Game,” Analysis 54:4 [October 1994], 196-200.)

Choose a number on this clock face and, starting from 12, spell out that number’s English name as you advance clockwise around the face, one letter per numeral. For example, if you’ve chosen 3, count out T-H-R-E-E and you’ll land on the numeral 5. Adopt this new position as your next chosen number and proceed as before (in this case, counting F-I-V-E and landing on 9). After three or more moves you’ll reliably land on 1.

This works because of a characteristic of Markov chains first observed by Russian mathematician Evgenii Borisovich Dynkin. Here’s a card trick that exploits the same principle.

This recording presents four pairs of tones, each pair separated by three whole tones, or half an octave. Curiously, some listeners hear the interval as ascending, others as descending. (In fact the tones used are ambiguous as to octave, so there’s no objectively right answer.)

Even more curiously, sometimes a listener’s perception reverses when an interval is transposed, say from C-F# to G#-D, even though nothing else has changed.

Diana Deutsch discovered the effect in 1986.

A pleasing detail from Built for Speed, University of Idaho zoologist John A. Byers’ 2003 account of a year studying pronghorn antelope in western Montana:

“It can be really bizarre if, when I’ve been alone for several hours, I spot a human standing or walking in the distance. For an instant, my reaction is, ‘What the hell is that?’ A large mammal that walks on its hind legs seems very strange.”