The Jan 1, 2012 issue of The College Mathematics Journal featured articles inspired by the late-great Martin Gardner. One by Ian Stewart begins with the Three Penny Puzzle invented by Gardner and Karl Fulves:
Gardner’s three-penny trick:Not being very good at card tricks and such, I would sometimes perform this trick in study hall or odd class moments to amuse my students, and then challenge them to duplicate the trick. I avoided doing it more than once in order not to have them catch on to the fact that the same pattern always works (why children). In all the many times I did it, and sometimes had students discover the logic, I don't think I ever asked them if they could do the same with four pennies, or five, or n. That is, for n Pennies, what is the minimal number of blind-flips f(n) to get them all alike, and what is the sequence of flips.
The trick is performed by a blindfolded magician. A volunteer places three pennies
in a row, and chooses at will whether each coin shows heads or tails. However, both heads and tails must appear, otherwise the trick ends before it begins. The magician announces that even though she cannot see the coins, she will give instructions to turn coins over so that all three coins show the same face, heads or tails.
The instructions are:
1. Flip the left-hand coin.
2. Flip the middle coin.
3. Flip the left-hand coin.
After steps 1 and 2 the magician asks whether all three coins show the same face,
and if the answer is ‘yes’, the trick stops, otherwise the magician requests the third flip.
Although it is plausible that enough flips will eventually
get all coins the same way up, it is a little surprising that at most three flips are needed.
I assume that the questions from the solver can only be "are all coins alike now?"
This is a great problem! A decade or so ago, I ran across a variant in which the position of the coins can be changed in between steps, but your commands can consist of more than one coin to be flipped. These are the "cup flipping" problems in my invariants handout at <a href="http://www.marinmathcircle.org/handouts/20102011/one%20player%20games%20with%20invariants.pdf>Marin Math Circle</a> for example.
ReplyDeleteThere's a whole family of problems here: for what numbers of cups, and what allowed permutations, can you guarantee a win in finitely many moves? The problems I gave here only allow rotations.
I think in your coin problem, if you only allow rotations around the center penny, you can still solve it (as long as your commands can include flipping more than one coin).