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Two Burning Ropes Timer Riddle

Two Burning Ropes Timer Riddle

The burning ropes timer riddle

Measure a time of 45 minutes by burning two ropes each of which takes 1 hour to burn out.

The ropes are of same length but none of the two burns at a uniform rate throughout the length. That's why you cannot just measure a point at three-fourth of the length of a rope and start burning at that point to measure 45 minutes.

Only saving grace is the allowance that one or both of the ropes can be burned simultaneously at multiple points.

Time to solve: 15 minutes.

Solution to the two burning ropes timer riddle: Step by step reasoning

From the information, draw the first key conclusion on what cannot be done,

Conclusion 1: As the ropes burn at non-uniform rates along the length, each rope can be burned only from the ends.

On further analysis, make the second key conclusion,

Conclusion 2: As burning a rope from only one end will burn it in one hour, target time of 45 minutes will be exceeded. So, no rope can be burned from only one end.

From the two conclusions, decide the first action,

Conclusion 3 on action Step 1: Burn one rope from both ends simultaneously. It will double the speed of burning. The rope will burn in 30 minutes, not in 1 hour.

This is your first breakthrough. A time less than 45 minutes is measured in the first step of burning one rope.

Challenge is now, how to burn the unused second rope to measure 15 minutes more.

Consider the two possible actions,

  1. If the second rope burns from both ends just AFTER the first rope burns out, it'll take 30 minutes more to burn out. That'll be 1 hour total. Won't work.
  2. If the second rope burns from both ends just as the first rope starts to burn from both ends, both ropes will burn out in 30 minutes. Won't work again.

In both infeasible actions, the second rope burns from both ends. So,

Conclusion 4: The second rope must burn only from one end.

Conclusion 5: Burning of the second rope cannot start AFTER the first rope burns out. No way 45 minutes target can then be met.

Conclusion 6 on action Step 2: Burn the second rope only at one end simultaneous to burning the first rope at both ends.

This is the second breakthrough achieved by step by step reasoning.

But solution not yet reached. So continue to reason based on knowledge gained till now.

Conclusion 7: As a rope cannot be burned only from one end, the second rope must be burned from its other end as well.

When should we light up the second rope at its other end?

Fact: The only known point of time to burn the other end is when the first rope burns out.

Conclusion 8: This is when we must start to burn the other end of the second rope.

Though reason says, there cannot be any other way 45 minutes can be measured, we are not satisfied.

This is an intriguing riddle where the solution is reached by finding the ONLY FEASIBLE course of action, though we do not know yet how 45 minutes will elapse after the actions are taken.

To be sure, let us examine burning the second rope in more details. What happens when the second rope starts to burn first time from one end simultaneous to burning first rope from both ends?

  1. The first rope burns out.
  2. 30 minutes of time elapses,
  3. 30 minutes half-life of the second rope and NOT ITS HALF LENGTH burns out. Half the full burn time of 1 hour of the second rope is spent. 30 minutes burn time of second rope still left.
  4. The burning point is still alive in the second rope and is moving towards its opposite end.
  5. In 30 minutes more, rest of the second rope will be burned out.

Significance of burning the second rope from the other end just as the first rope burns out,

Result of last action: Other end of the second rope lighted up just as the first rope burns out in 30 minutes. This is same as burning the rest half-life of the second rope from both ends simultaneously. Burning 30 minutes rest half-life of the second rope in double the burning speed of one end burning, takes 15 minutes. Both ropes burn out in $30 + 15 = 45$ minutes.

This is how 45 minutes of time is measured.

Let's summarize the solution by collecting the steps together.

Solution step 1: Start burning the first rope from both ends and the second rope from one end simultaneously.

Solution step 2: Start burning the second rope from the other end just as the first rope burns out fully in 30 minutes.

The problem is solved step by reasoned step.

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