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## 4 buttons and a light bulb 1

There are two rooms, in one room there is a light bulb, and in the other there is a round table, with 4 buttons arragned in a square on it. Each button may be on or off, and the light bulb will be on only if all the buttons are on. Every time you leave the buttons room to the light bulb room, the table with the buttons rotates randomly. How, without knowing the buttons' initial state, you can make sure the light bulb will turn on?

notes:

• The buttons all look the same, and you can't distinguish between them.
• You can't distinguish between a turned on button and a turned off button, the only way to gain information on the buttons' state is the light bulb.
• You can't see the light bulb from the buttons room, you need to enter the light bulb room to see it.
• The light bulb doesn't produce any heat, so you can't feel if it was on before by touching it.
see solution

Hint:

We can't keep track of the buttons' state (it changes every time since the table rotates). So, we will divide all the states to six groups, with the condition that in each group, if you rotate a state, you will get another state of the same group. (If you are familiar with equivilance relations, we are dividing the states to equivilance classes, with the relation "x can be rotated in order to get y"). These are the groups we get:

• A - all the buttons are off
• B - exactly one button is on
• C - two buttons are on, and they are near each other
• D - two buttons are on, and they are opposite to each other
• E - three buttons are on
• F - four buttons are on

We know that if previously you had a state in group X, when you go to the light bulb and return, it will still be an X state, since the states in X can rotate to each other. So, we can keep track of the groups of states, instead of the states themselves.

Solution:

We can eliminate those groups, one at a time, while keeping track of the current possible state-groups at each step:

• We start with either A, B, C, D, E or F (all possible states).
• 1 - Eliminate F: Check if the light is on, if it is your'e done, if not, the state can't be in F, we remain with A, B, C, D, E.
• 2 - Eliminate A:
• 2.1 - Press all the buttons, A will turn into F, B will turn into E, C to C, D to D, and E to B. now we have B, C, D, E, F.
• 2.2 - Eliminate F: Check if the light is on. if not, we remain with B, C, D, E.
• 3 - Eliminate D:
• 3.1 - Press two buttons that are opposite to each other, B or E will remain B or E (they have odd number of buttons, and by pressing two buttons you remain with and odd number), C will remain C, and D will become either A or F. we now have A, B, C, E, F.
• 3.2 - Eliminate A and F (like steps 1, 2): check if the light is on, if not, press all the buttons and check again. we now have B, C, E.
• 4 - Eliminate C:
• 4.1- Press two buttons that are near each other, B or E will remain B or E like before, C will become either A, D, or F. we now have A, B, D, E, F.
• 4.2 - Eliminate A and F: check if the light is on, if not, press all the buttons and check again. we now have B, D, E.
• 4.3 - Eliminate D like step 3:
• 4.3.1 - Press two opposite buttons, B, E will remain B, E, and D will become A or F. we have A, B, E ,F.
• 4.3.2 - Eliminate A and F: check if the light is on, if not, press all the buttons and check again. we now have B, E.
• 5 - Press one button, B will become A, C, or D, and E will become C, D, or F. we have A, C, D, F.
• 6 - Eliminate A, F: check if the light is on, if not, press all the buttons and check again. we now have C, D.
• 7 - Eliminate D:
• 7.1 - Press two buttons that are opposite to each other, C will remain C, and D will become either A or F. we now have A, C, F.
• 7.2 - Eliminate A and F: check if the light is on, if not, press all the buttons and check again. we now have a C state.
• 8 - Eliminate C:
• 8.1- Press two buttons that are near each other, C will become either A, D, or F. we now have A, D, F.
• 8.2 - Eliminate A and F: check if the light is on, if not, press all the buttons and check again. we now have a D state.
• 8.3 - Eliminate D:
• 8.3.1 - Press two opposite buttons, D will become A or F, so we have A, F.
• 8.3.2 - Eliminate F: check if the light is on, if not, we must have an A. now we can press all the buttons to get an F, and wer'e done!

faizan

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leilei3915

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