Imagine you are outside a closed room with no windows and
one door. You cannot see inside this room until you open the door.
There are three light switches in front of you. Each switch is
connected to a different one of three light bulbs inside the closed room.
All the light bulbs are currently off and thus all three switches are
currently in the off position. You may flick on and off any of these
switches as much as you like UNTIL you open the door and go inside
the room to witness the state of the light bulbs. At that point you may
no longer change the switches.
You may only open the door ONCE. THEN you must be able to
tell which switch is connected to which light bulb.
In this scenario, how can you, with 100% certainty, tell which
switch goes to which light bulb?
The Burning String Problem:
You have two lengths of string, which although they are not the
same length, will each take exactly 30 minutes for a flame to burn from
one end of the string to the other. The strings do not burn evenly, so the
position of a flame, as it burns through a string, will not afford you any
specific knowledge of how much time the string has been burning for.
All you know is that it takes exactly 30 minutes for a flame to burn
from one end of each string to the other end.
Without using a timepiece, how can you use these two strings
and matches to measure out exactly 45 minutes with almost perfect
accuracy?
two different ways
Solution 1: Turn on one switch (call it 'switch one ) and let it
stay on for a while. Then turn that 'switch one off and turn on
another switch (call it 'switch two ). Open the door to the closed room
and walk inside. The lit light bulb is obviously connected to 'switch
two . Now touch the two unlit light bulbs. The hot unlit light bulb is
connected to 'switch one . The cooler unlit light bulb is connected to
the switch you never turned on.
Solution 2: Turn on one switch (call it 'switch one ) and let it
stay on for a year. Then come back and turn on another switch (call it
'switch two ). Open the door to the closed room and walk inside. The
lit light bulb is obviously connected to 'switch two . Now examine the
two unlit light bulbs. The burned-out light bulb is connected to
'switch one . The still fresh unlit light bulb is connected to the switch
you never turned on.
The Burning String Problem: Light one of the strings aflame,
not just on one end, but on both ends simultaneously. When the two
flames reach each other (wherever along the string that may be), you
will know you have measured out exactly 15 minutes. Immediately
light one end of the other string and allow that to burn all the way
through for the final 30 minutes. Voila! 45 minutes.
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