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[sitnaltax]

I enjoy collecting good logic riddles, and I thought some of y'all forumites might enjoy them as well.

No "lateral thinking" here--there's no hiding bad assumptions in the problem description or flavor and no tricking your way around them.

I'll start with what I think is my very favorite:

An evil overlord has captured you and, as evil overlords are wont to do, he forces you to play a game to win your freedom.

The overlord shows you a turntable with four bowls on it, one at each compass point. You are blindfolded and the bowls are set to some unknown orientation: some face up, others face down. Each round of the game works like this: You put your hands above two of the compass points. The overlord then rotates the turntable in any way he wishes (but always stopping at the compass points; there are only four possible orientations for it). You then lower your hands onto the bowls that are now beneath your hands and are allowed to flip both, one, or neither of them. If all four of the bowls are now in the same orientation--either all face up or all face down--the overlord announces your victory, you immediately win and go free. Otherwise you play another round; however, after 10 rounds, if you still haven't won, the overlord will send you to a horrible fate. The overlord wants this to happen, of course, so he will use his ability to rotate the turntable to thwart you if possible.

What strategy can you use to guarantee victory?

Hint 0: A solution exists.
Hint 1: 10 rounds is far more than necessary. In the worst case, you can do it in 7 rounds. As the faithful, perspicacious, and virtuous followers of this thread have noticed, my original clue was incorrect.
Hint 2: You don't need to realize that you've won to win.
Hint 3: Amazingly, the solution also works if the bowls are replaced with coins, and you are wearing gloves so you can't determine heads from tails.
Hint 4: How must the bowls be oriented in order to force a victory on your next move? How can you cause that situation to exist?

As a postscript, Hint 3 is one of my favorite hints of any riddle, because at first glance it only seems to make the problem harder.

[pacovf]

I assume you know the orientation of the "chosen" bowls before you decide whether you want to flip them?

[sitnaltax]

I assume you know the orientation of the "chosen" bowls before you decide whether you want to flip them?

Yes.

[enfynet]

But, if you are lowering your hand to the bowls before deciding, wouldn't you just pick the one that was flipped the wrong way? It could be over in two turns. (Depending on the first two table stops)

[Jimmmmm]

Without looking at the clues:

1. Choose opposite bowls. Set them both down.
2. Choose opposite bowls again. If they're not both down, set them both down and win. Otherwise, set them both up.

You now know you've flipped the same bowls twice, otherwise you would have won after turn 1. The 2 bowls you haven't already flipped will be different to each other, otherwise you would have won after turn 1 or 2. So 1 is down and 3 are up.

3. Choose adjacent bowls. If 1 is down, set it up and win. Otherwise they'll both be up; set one down.

Now you'll have 2 up and 2 down, either both pairs of opposite bowls will be the same, or 2 pairs of adjacent bowls will be the same.

4. Choose opposite bowls. If they are the same, flip them both and win. If they are different, leave them.

Now you'll have 2 adjacent up and 2 adjacent down.

5. Choose adjacent bowls. If they are the same, flip them both over and win. If they are different, still flip them both over.

Now you'll have 2 opposite up and 2 opposite down.

6. Choose opposite bowls, flip them both and win.

[werothegreat]

After reading hint 3:

0) Flip nothing.  If you don't win, either opposites match, or adjacents match, or three match.

1) Choose opposite.  Flip them both over.  If you don't win, either adjacents match, or three match.

2) You know opposites don't match.  Choose adjacent and flip both over.  If you don't win, either opposites match, or three match.

3) Choose opposite.  Flip them both over.  If you don't win, three match.

4) Stay opposite, or choose adjacent - it doesn't matter which.  Flip one over.  If you don't win, that means either adjacents or opposites match.

5) Choose opposite.  Flip both over.  If you don't win, that means adjacents match.

6) Choose adjacent.  Flip both over.  If you don't win, that means the matches were perpendicular to you, and you just made opposites match.

7) Choose opposite.  Flip both over.  You win.

This works with the coins and the gloves, but I'm not sure how you can get it down to 5 steps.

[sitnaltax]

Jimmmmmm and werothegreat have it--very nice. Wero, you have the interesting part, although there's a minor efficiency to be eked out. I made an error in Hint 1 and have fixed it--apologies. Jimmm's answer is (I believe) the most efficient and is the same as mine except for the part where he counts correctly.

Enfynet, the overlord is an adversary, and will spin the turntable to keep you from winning if possible. For instance, say in round 1 you flip two from down to up. Now you know there are 3 up/1 down (because you haven't won yet). On round 2, the overlord will definitely spin the turntable so the remaining down bowl is not under either of your hands.

[eHalcyon]

Jim's solution doesn't work for coins.  I don't see how wero's can be more efficient.

I got the same solution as wero except that I start by flipping opposite twice in a row.  It works out the same, though I thought wero snuck extra efficiency somehow until I noticed he numbered starting from 0.

[GendoIkari]

I'm pretty bad at solving these, but I love reading about them and presenting them. I was about to post my favorite, but I seemed to recall doing so already. Found it:

http://forum.dominionstrategy.com/index.php?topic=6426.0

[Jimmmmm]

I'm pretty bad at solving these, but I love reading about them and presenting them. I was about to post my favorite, but I seemed to recall doing so already. Found it:

http://forum.dominionstrategy.com/index.php?topic=6426.0

Yeah, that's a cool one.

[Grujah]

We had this thread before.
http://forum.dominionstrategy.com/index.php?topic=6426.50


I think I've got it.
Got my solution after hint 2, but I didn't really need them. I don't think it works with hint 3 variation, though:



1) Choose adjacent - Turn both down.
2) Choose opposite - Turn both down. Now you know there are 3 down and one up.
3) Choose adjacent - You will get two down bowls. Flip one up. Now you have a 2/2 split - either opposites or adjacent match. Its 50/50.
4) Choose opposite - If you get the same, flip them, you win. Otherwise, don't flip them - in that case, adjacent match.
5) Choose adjacent - Flip them. Either you won or opposite match now.
6) Choose opposite - Flip them. win.


now that I've read my solution in writing, I can probably fiddle with first few moves to make sure it fits hint 3 as well.

I see that except first two steps my solution is same as Jimmm's, I just took different approach to getting 3 down and 1 up.

[Avin]

For the coin/gloves variant, wero is doing two extra steps.


1) Do nothing
2) Flip opposite
3) Flip adjacent
4) Flip one
5) Flip opposite
6) Flip adjacent

I made a little graph with four nodes on it (labeled 4, 3/1, 2A and 2O)  and edges marked with the type of transition to get from one node to the other. The above sequence of 6 moves guarantees that you end on the 4 node at least once.

2A represents the configuration where you've got 2 heads at adjacent directions and 2 tails on the other directions; 2O represents the configuration where you've got 2 heads on opposite directions and 2 tails on the others.

You have to do nothing to start with because if you start in a solved state, then the remaining moves will never get you back there. Step 2 solves you if you started in 2O and effectively does nothing if you were in 2A or 3/1. Step 3 solves you if you started in 2A, and also does nothing if you started in 3/1. At this point the only configuration left to solve is 3/1, and flipping one coin in step 4 moves you out of 3/1 but you have no way to determine which state you landed in (unless you immediately win), so repeat steps 2 and 3 to solve.


I enjoyed this, and I've also never heard it before, which is surprising since I also enjoy collecting good logic puzzles!

[werothegreat]

The problem, Avin is that you can never win by flipping adjacent because the overlord can always move the turntable so that you don't flip the right ones.

[Avin]

You're right, I left off an edge in my graph.

[Avin]

So I think we can skip the first empty step we both have though since the problem statement says "some face up, others face down." That brings it down to 7 moves for the coin problem.

I don't see that it's possible to get down to 6 though without making any assumptions about the behavior of the overlord. For instance, if we are in a 3/1 state (e.g. after wero's step 3) and we select opposite corners, if we can assume that the overlord will NOT allow the 1 to be under our hand in order to prevent us from immediately escaping, then we can flip one and know that we ended up in the 2A state, so we would then need to just flip adjacent and opposite. That would get it down to 6 moves, if that assumption about the overlord is justified, which I don't think it is.

[eHalcyon]

In puzzles like these, "some" usually means that 0 is a possibility as well.

[Avin]

In puzzles like these, "some" usually means that 0 is a possibility as well.

This statement seems to indicate otherwise:

For instance, say in round 1 you flip two from down to up. Now you know there are 3 up/1 down (because you haven't won yet).

You couldn't "know" that there are 3 up / 1 down in the formulation of the puzzle that "some" includes the possibility of 0, because starting with 4 down would be a valid scenario.

Not spoilering this because I don't think discussion of the original rules is spoilery.

[eHalcyon]

In puzzles like these, "some" usually means that 0 is a possibility as well.

This statement seems to indicate otherwise:

For instance, say in round 1 you flip two from down to up. Now you know there are 3 up/1 down (because you haven't won yet).

You couldn't "know" that there are 3 up / 1 down in the formulation of the puzzle that "some" includes the possibility of 0, because starting with 4 down would be a valid scenario.

Not spoilering this because I don't think discussion of the original rules is spoilery.

Eh, maybe not in this puzzle then.  But it's not an assumption that should be made for most logic puzzles.  If there must be at least 1 up and 1 down to start, it should be explicitly stated.

[Jimmmmm]

For the coins (basically a summary of what others have said):

There are 4 possible states of the bowls:  3 the same (3), adjacent pairs the same (A), opposite pairs the same (O), and all 4 the same (4).
We have 3 possible moves we can make: flip opposites (o), flip adjacents (a) and flip 1 (1).

From O, o is a winning move.
From A, ao is a winning move (since a takes A to O).
From either O or A, oao is a winning move (since o takes A to A).
From 3, 1oao is a winning move (since 1 takes 3 to either O or A).
Therefore, from O, A or 3, oao1oao is a winning move (since o and a take 3 to 3).

If it's possible to start in state 4, add any move to the start (it can be the zero move, but 1 might be a good choice, since 3 seems the most likely starting state).

[werothegreat]

For the coins (basically a summary of what others have said):

There are 4 possible states of the bowls:  3 the same (3), adjacent pairs the same (A), opposite pairs the same (O), and all 4 the same (4).
We have 3 possible moves we can make: flip opposites (o), flip adjacents (a) and flip 1 (1).

From O, o is a winning move.
From A, ao is a winning move (since a takes A to O).
From either O or A, oao is a winning move (since o takes A to A).
From 3, 1oao is a winning move (since 1 takes 3 to either O or A).
Therefore, from O, A or 3, oao1oao is a winning move (since o and a take 3 to 3).

If it's possible to start in state 4, add any move to the start (it can be the zero move, but 1 might be a good choice, since 3 seems the most likely starting state).

i.e., it's a solution with 8 moves, and sitnaltax needs to fess up what his "better" solution is.

[sitnaltax]

For the coins (basically a summary of what others have said):

There are 4 possible states of the bowls:  3 the same (3), adjacent pairs the same (A), opposite pairs the same (O), and all 4 the same (4).
We have 3 possible moves we can make: flip opposites (o), flip adjacents (a) and flip 1 (1).

From O, o is a winning move.
From A, ao is a winning move (since a takes A to O).
From either O or A, oao is a winning move (since o takes A to A).
From 3, 1oao is a winning move (since 1 takes 3 to either O or A).
Therefore, from O, A or 3, oao1oao is a winning move (since o and a take 3 to 3).

If it's possible to start in state 4, add any move to the start (it can be the zero move, but 1 might be a good choice, since 3 seems the most likely starting state).

i.e., it's a solution with 8 moves, and sitnaltax needs to fess up what his "better" solution is.

Sure. Fingers crossed that I didn't screw anything up. I like Avin's notation for the possible states, so I'll stick with that.
Although I realize I wasn't as crystal clear as I could have been in the description, I consider it an assumption that the initial state is nonwinning. If you don't like that, then yeah, you have to add a do-nothing "step 0" at the beginning.

1. Flip opposite. Now I am in 2A or 3/1.
2. Flip adjacent. Now I am in 2O or 3/1.
3. Flip opposite. Now I am in 3/1.
...
...
3. Flip one. Now I am in 2A or 2O.
4. Flip opposite. Now I am in 2A.
5. Flip adjacent. Now I am in 2O.
6. Flip opposite.

I have used the number 3 twice to reflect the fact that I also used that number twice when counting for the clue, apparently. My shame is immense.

[sitnaltax]

I'm pretty bad at solving these, but I love reading about them and presenting them. I was about to post my favorite, but I seemed to recall doing so already. Found it:

http://forum.dominionstrategy.com/index.php?topic=6426.0

Damn. I even looked (searched for "riddles", because I quickly found sudgy's recent post about pen-and-paper logic puzzles) but didn't go back this far. Good stuff there, including some of the ones I had meant to post.

My favorite follow-on to the blue eyes/brown eyes problem, although one that I don't have a good handle on myself: Why does the Guru making a statement that everyone already knows is true matter? If he had come in and said "Bacon is delicious", would that have had the same effect? Why not, given that it added no new information?

[popsofctown]

Can't the overlord just always win if North is flipped up, South is flipped down, he starts off by letting you mess with East and West and always rotates the table 180 degrees or 360 degrees?

Are only 90 and 270 allowed for the overlord? If so I think you should really specify in the problem description, it seems about as relevant as 45 being forbidden, which was made clear.

[Grujah]

No, he can rotate 90, 180, 270 or 360 (0).
You place hands, than he rotates. You can place your hands on adjacent positions (North + East, for example) or opposite ones (North + South).

[werothegreat]

I have used the number 3 twice to reflect the fact that I also used that number twice when counting for the clue, apparently. My shame is immense.

So that means I got it completely right!