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Let's say that you are setting the dining table for a party of 60 guests. The 60 guests will be seated at a giant circular table. Each seat has a lamp, and currently some of the 60 lamps are on and some are off. The butler offers to play a game with you. Let's say that the chairs are labelled with the guests names, which for convenience we label starting from the top and going clockwise as C1, C2, ..., C60, and the chairs are stationary. The giant circular dining table along with the lamps, however, can rotate. Each lamp has a switch, and flipping the switch when the lamp is on turns it off, and vice versa. The game is as follows

1) The game consist of turns. At each turn, the butler tells you a list of chairs at which he want to switch the lamps' state.

2) You will then has the option of arbitrarily rotating the entire table before applying the switches at the list of chairs in point 1). Please note that the you must rotate the table in such a way that each chair is directly across from a lamp (rotating so that a lamp ends up midway between two chairs isn't allowed). You can turn the table in any direction and for as many spaces as you want before applying the switches.

3) You may take as many turns as you'd like. If the butler succeeds in turning all the lights on, he wins.

Show that under certain starting lamp patterns, there is a strategy which guarantees that the butler will never be able to turn all the lights on.

EDIT: added the proper initial constraints thanks to superprimastic and witzar.

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There is at least one initial state of

the lamps where it is impossible for

you to stop him from turning them all

on. Consider the case where the lamps

start off alternating on then off.

No mater how you rotate the table, the

lamps will get to be all on (the butler

wins) or all off, In the latter case,

the butler's next move is to list all

the chairs so that all will be turned

on no matter how you rotate the table.

I haven't checked, but there may be

other such anomalous cases.

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There is at least one initial state of

the lamps where it is impossible for

you to stop him from turning them all

on. Consider the case where the lamps

start off alternating on then off.

No mater how you rotate the table, the

lamps will get to be all on (the butler

wins) or all off, In the latter case,

the butler's next move is to list all

the chairs so that all will be turned

on no matter how you rotate the table.

I haven't checked, but there may be

other such anomalous cases.

Thanks superprismatic and witzar. That is really a silly oversight in the wording of the puzzle. The intent is to show that under certain starting lamp patterns, there is a strategy which will guarantee that the butler will never be able to turn all the lights on. I'll fix it in the OP. Thanks for pointing that out.

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