Comments on: A double flip

By: Greg

Greg — Fri, 19 Mar 2010 23:00:21 +0000

It shouldn’t be possible to find a “golden rule” to handle all possible mattress situations no matter how many mattresses you add: this would require the group of symmetries to be a cyclic group, but the group needs to have the symmetries of any single mattresses as a subgroup. The group of symmetries of the single mattress has three distinct elements of order two, which is impossible in a cyclic group.

By: John Cowan

John Cowan — Thu, 18 Mar 2010 00:33:09 +0000

Unless you have taken the law into your own hands and removed the tag, the mattress is marked already. So if you confine yourself to rolling and yawing, both of which flip the side the tag is on, you can simply identify one side of the bed as the roll side and the other as the yaw side.

By: Jon McAuliffe

Jon McAuliffe — Wed, 17 Mar 2010 23:41:54 +0000

My solution to this is to toss a coin each time. Heads is a P (flip over the head), tails is an R. I flip mine everytime I change the valance though, (significantly more than once a season).

By: Nemo

Nemo — Wed, 17 Mar 2010 19:45:36 +0000

P.S. As for the Hilbert Hotel, my rule would be to move the mattress in room N to room N+1 for each N. If N is odd, roll the mattress; if N is even, flip the mattress.

Would my rule count as constant mental effort, or infinite mental effort?

Of course, you will need to keep adding new mattresses. The Hilbert is like that.

By: Nemo

Nemo — Wed, 17 Mar 2010 19:24:50 +0000

If you can alternate rolls and flips (say), you can arrange for any single mattress to have all of its orientations visited.

I do not understand Tim Knoll’s description, exactly… But his idea is sound. If you simply have a “ring” of two or more beds, and you roll each mattress as it enters bed 1 and flip it as it exits bed 1, then obviously each mattress will visit every possible orientation (because its own sequence of rolls and flips will alternate as it keeps passing through bed 1).

However, if you consider the combination of mattress orientation and bed number as the “state” of the mattress, then at least half of the total states will never be visited. In my example, for instance, “mattress A on bed 1″ will only be seen in half of the possible orientations. I do not believe this can be fixed without introducing memory into the protocol (or marking the mattresses).

This claim is not too hard to prove… Without memory, any rule involving moving mattresses among multiple beds reduces to a set of disjoint “rings” (cycles), where each bed has a fixed “next bed” and a fixed “next mattress operation” (I, R, P, or Y). Again, if you focus on a single mattress and a single bed, you can easily show it can only reach half of the possible orientations on that bed.

By: Paul Topping

Paul Topping — Wed, 17 Mar 2010 17:47:13 +0000

Not sure about multiple beds but I thought the flipping scheme was pretty simple for a single bed. There are only 4 possible places for ones head to go on a two-sided, two-ended bed. Just mark each of these with the names of the seasons and when each season starts place the end with its name where you would like your head to be. If you miss a season, it all gets back in sync next season. Save your group theory for a tougher problem. ;-)