You can look also at more difficult version of this puzzle with 9 coins.
Heads – tails
As it often happens in life evil warlock imprisoned a wise man. Surprisingly he gave him a chance to save himself if he fulfils one task.
There is a round tray on the table which can be turned around freely. There are four coins on the tray placed as corners of a square.
Wiseman has his eyes covered so he doesn’t see anything. His task is to turn the coins so that they are all heads up.
He is not in easy situation. He may turn some coins and than the warlock will turn the tray around. He turns some coins again warlock turns the tray again. It goes on until all coins are turned correctly. At that moment evil warlock ends the game.
Wiseman can’t recognize one side from the other by touching it. He has to leave the coins where they were in the corners of a square. And mainly – because of the evil magic – he has the worst luck so if he relies just on chance he will never fulfill the task.
Hint
Wiseman will have to proceed somewhat symmetrically so that he wouldn’t mind that warlock is turning the tray around. Only thing that he can rely on is whether the warlock has ended the game.
Answer
Let’s make the puzzle a little easier first: let’s say it’s enough if all coins are the same side up. It doesn’t matter if heads or tails.
If all coins are turned same side up we are done.
Wiseman will assume that two coins are heads up and two coins are tails up.
He turns two coins in opposite corners. If his assumption was correct and two same side up turned coins were not next to each other (were across from each other) warlock will end the game now.
If it did not happen wise man will not give up his assumption about two + two coins yet. If two coins weren’t across from each other they have to be next to each other (the previous turn didn’t change it).
Wiseman will turn two random coins placed next to each other which should (if the 2+2 assumption is correct) either end the game or the coins will be across from each other. If it didn’t end the game wise man will simply turn the coins in opposite corners and that should end the game.
Game is still not over? It seems that wise man's assumption about two and two coins was wrong. It means that one coin is facing up different side than others. Now it’s easy. Wiseman turns any coin. Now they are either all the same or two and two.
It means that wise man will repeat what was described above. Now he is sure that there is heads on two coins and tails on the other two.
Let’s get back to the original puzzle. It is a bit more complicated but just a little bit. Wiseman does the same as described above only his every other turn will be turning all four coins.
I hope I explained it clearly enough.
