Calculating 25 minutes by burning 2 inconsistently burning 1-hour ropes
You are given two ropes and are told that they burn at inconsistent rates, but will always take 1 hour to completely burn up. This means that cutting one of the ropes perfectly in half will not give you two smaller 30-minute ropes.
You are told that you need to approximately calculate 25 minutes by burning these ropes in some fashion.
How do you accomplish this?
Hint:
first ½ of hint
I carefully chose approximately instead of precisely
second ½ of hint
because to calculate it precisely would require that you ignite an infinite number of flames
Answer context: if you light both ends of one rope, the two flames will always meet each other and fizzle out in exactly 30 minutes.
Key takeaway:
So you may (correctly) think that 3 flames = ⅓ hour. But there’s trick to it. Those 3 flames should always be; 2 on the end, and 1 arbitrarily in the middle.
Doing this will actually cause 1 rope segment with 3 flames to turn into 2 rope segments with 4 flames. Since that middle flame was placed arbitrarily, we can expect one of the segments to burn up before the other. When this happens, we just place a new flame arbitrarily in the middle of the remaining segment, splitting it back into 2.
As long as we keep adding flames so that we always have 2 segments, the total rope will ultimately burn in 20 minutes (⅓ hour).
This is generalizable; to achieve any fraction of 1/x hours, we need to always maintain x-1 segments (min 1). So for ½ hour, we need to maintain 1 segment. For ⅐ hour, we need to maintain 6 segments, etc.
Final answer: 25 minutes can be expressed as ⅙+¼ hours. Therefore we need to burn the first rope such that we always maintain 5 segments. Then once that’s fully burnt, we just burn the second rope such that we always maintain 3 segments.