**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.

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.

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.

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 ?

AVERAGE is 3

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.

I like this brain teaser…

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.

The answer tends to 3.

Could you elaborate how did you do it in Excel

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.