In my November Diary I grumbled at length about all the Diversity-Inclusion-Equity (DIE) flapdoodle that increasingly blights the Notices of the American Mathematical Society. Along the way I mentioned an article in the Notices issue I was looking at, the issue dated December 2019. The title of the article was "Talent Nurturing in Hungary: The Pósa Weekend Camps." My mention of this article was collateral damage, not directly relevant to my grumbling; the article is not seriously DIE-blighted. (And I was reading the online version of the Notices; this article does not appear in the print version.)
Having mentioned the article and skim-read it, I thought I'd pluck a brainteaser out of it. This one:
A precious piece of treasure is locked up in a safe. The door of the safe is circular and there are four indentations on it. The indentations are positioned on the vertices of a square centered at the midpoint of the circular door. Each indentation hides a binary switch, which cannot be seen from the outside but its position can be identified if we put our hands in.
We are standing in front of the door and we can each put our hands into one of the indentations, and we can change their setting. However, once we pull our hands out, the door senses that it has been tampered with, and it starts rotating extremely fast until it stops at possibly a different angle than before. Unless of course all four switches are in the same position, in which case the door opens immediately and we win the treasure. Can we always win the treasure in finite time?
When, a few days later, I tackled the problem, I concluded that it is a dud. Possibly I'm missing something.
First off, who are "we"? We were told in the main text of the article that:
The students … work in teams of two, three, or sometimes four.
If there are four of us, we can all put our hands in simultaneously and set the four switches to the same position. Door opens, we win.
If just one person puts in a hand and flips a switch, there's a probability p less than 1 his flip will not open the door. He withdraws his hand, the door rotates. After N repeats of this, there's a probability of the order pN the door is still not open. This probability will get tiny as N gets large, but it will not necessarily be zero. Similar remarks apply if two or three people are flipping simultaneously.
Am I missing something? Or did the problem lose something in being translated from Hungarian?