# 3 Doors and Heaven

Problem:

A person dies, and he arrives at the gate to heaven. There are 3 doors in the heaven. One of the door leads to heaven, second one leads to a 1-day stay at hell and then back to the gate and the third one leads to a 2 day stay at hell and then back to the gate. Every time the person is back at the gate, the 3 doors are reshuffled. How long will it take the person to reach heaven?

Solution:

According to probability 1/3 of the time, the door to heaven will be chosen, so 1/3 of the time it will take 0 days. 1/3 of the time, the 1-day door is chosen, of those, the right door will be chosen the next day. Similarly, 1/3 of the time, the 2 day door is chosen, of those, the right door will be chosen after the 2 days.

So lets say it will take N days. 1/3 of the cases are done in 0 days as before. 1/3 of the cases are 1+N. 1/3 are 2 + N.

N = 1/3 * 0 + 1/3 * (1 + N) + 1/3 * (2 + N)
N = 1 + 2N/3
Therefore, N/3 = 1 ; N = 3.

So it will take on average 3 days to reach to heaven.

### 0 Thoughts on “3 Doors and Heaven”

1. it should take 1 day, I think.
Lets say 3N people went. N were sent straight away, N were sent to hell for a day and the rest N were sent to hell for 2 days. So in the first day, N of 3N people went to heaven. add another N for the next day. keep repeating the process and take the average. we find that N people from 3N total reach heaven daily. thus, on average a person taken 1 day i.e. 2 visits.
I think it is about average of 0, 1 and 2 days which comes out to be 1 day.

2. Anonymous on February 24, 2014 at 12:49 am said:

Incorrect answer as the problem states 3 gates with:
1 gate directly to haven
1 gate to 1 day in hell and back to the gate (NOT heaven after a day)
1 gate to 2 days in hell and back to the gate (NOT heaven after 2 days)

So, I don’t think there is any perfect answer to this question as every time the gates would again be shuffled and a person may end up getting into the same gate again and again.

3. Soumya Chaurasia on August 31, 2014 at 3:03 am said:

As the gates are reshuffled again n again , then a person may end up selecting the same gate after 1st n 2nd go , So how come 3 ?

4. Here the question is not complete.
When we say reshuffle, is it happening randomly or will it change its previous state and will only return to same state after completing all the state.

So if we assume certain things that it will shuffle its state and will only return only after completing other states, then answer will be 3.

1st Day : Guy selects the door and goes to hell for 2 days
3rd day : Guy selects the same door and goes to hell for 1 day
4th day : guy selects the same door and goes to heaven

So in 3 days he will reach to heaven for sure.

• Richa on June 3, 2017 at 3:43 pm said:

Your answer is not correct. Your comprehension of the question is similar to saying let’s freeze the gates but we do not know which gate is which, given this situation what is the maximum number of days it would take to get to heaven.
The heaven-gatekeeper probably does not like you, considering that he gave three gates two of out of which leads to hell, thus he would not compromise so much with his riddle for your convenience.

If we consider two variables, the answer, mathematically very simply arrives at 1.125 days

5. Dharini Kathir on November 3, 2014 at 12:14 pm said:

I like this brain teaser…

6. I did run the excel-based simulation generating the random choice by the chooser. Ran this experiment for 1000 times and ran such experiment 100 times.

7. Attempt at a non-algebraic explanation:

Because there is 1/3 chance of the person choosing the right door, he/she will have to attempt, on average, 3 times.

Each attempt has a punishment, and that punishment is
1/3 of the time 0days
1/3 of the time 1days
1/3 of the time 2days
so by 0(1/3)+1(1/3)+2(1/3) = 1
The punishment is, on average, 1 day.

So if the person returns to the gate 3 times, the punishment is 3 days, on average.

PS: This also works if the punishment was 0, 3, or 6 days. The punishment would be on average 1/3(0+3+6) = 3 days. So if the person returns to the gate 3 times, the punishment is 9 days, on average.