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This is a game that was re-created in a recreational math session.

You are given four glass cylinders closed on top and bottom , and the curved edge being made of glass , with an arrow painted inside. The cylinders are placed on the four corners of a table that can be turned by multiples of $90 ^\circ$ by a person(say,host). You are then blindfolded and then you need to select any two of the cylinders on the table and flip either one or both of them. After each move of flipping, the host would turn the table by any multiple of $90^\circ$ that he wishes to. You need to devise a winning strategy to win in a finite number of moves. A win is when all of the cylinders face one way. You cannot (obviously) feel the orientation of the arrow by senses apart from sight.

The problem I run into is I do not know how to start an argument of turning the cylinders if I cannot figure out the original orientation of the cylinders.

I tried to do it case by case,but it was not very fruitful, so I do not feel it would be of any benefit to add the stuff I have done at this point.

I would appreciate any help.

Anuj jha
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  • Can you clarify the starting position? EG If 3 cylinders had arrows in the same direction, then this would be an invariant and hence you can never win. – Calvin Lin Nov 01 '21 at 19:41
  • @CalvinLin I do not think my instructor mentioned the original orientation,he said- "you can do it in less than 20 moves and that is all I will say". The initial position is what I was having trouble with as well. – Anuj jha Nov 02 '21 at 02:43

1 Answers1

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As mentioned in Calvin Lin's comment, this problem is not solvable if exactly three cylinders face one way. Here are some hints for a solution when it is known that two arrows face one way (and the other two face the other way):

Hints: Let $X$ denote a turn where (any) two diagonally opposite cylinders are flipped, while $Y$ denote a turn where (any) two adjacent cylinders are flipped.

  • Consider doing $X$. If we are lucky, we win with this one turn, but what if we don't? What can we say about the arrow directions now?
  • After doing $X$ and not winning, consider doing $Y$. Again, we might be lucky and win this turn, but what if we do not win in this turn too? What can we say about the arrow directions now?
  • If we somehow do not win after doing $X$ followed by $Y$, all is not lost. In fact, we can guarantee we can win in the third turn. What do we do?

Remark: This is somewhat similar to four glasses puzzle. The differences are (1) the directions of the two chosen cylinders are revealed to the player, and then (2) the player can choose to flip both, either, or none of them.

Edit: If we are allowed to choose to flip only one of the two selected cylinders, we can solve the problem for all initial states.

Let's say we have done the above three steps, and we still have not won. Then we now have three arrows facing one way, and one arrow facing the other way. (Why?) Now, simply flip exactly one of the cylinders.

  • If we are lucky, we win the game.
  • If not, now we know two arrows point to one way and the other two point to the other way. So, we can now do the three steps again to win the game.
VTand
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  • Thanks for the hints! This was actually a follow-up problem after the Four Glasses Puzzle. – Anuj jha Nov 11 '21 at 07:01
  • I'm sorry, but the question has a slight correction that I clarified with my instructor. We need to choose any two,but which ones to flip is up to us i.e we can flip any one of the two selected,or both. – Anuj jha Nov 11 '21 at 09:07
  • Thanks; that clarification helps to solve all cases of the initial arrow directions. – VTand Nov 11 '21 at 11:25