Maze Challenge For a Rat?

A rat is placed at the beginning of a maze and must make it to the end. There are four paths at the start that he has an equal chance of taking: path A takes 5 minutes and leads to the end, path B takes 8 minutes and leads to the start, path C takes 3 minutes and leads to the end, and path D takes 2 minutes and leads to the start.

What is the expected amount of time it will take for the rat to finish the maze?

This could be the average time that rat needed!
 

A Rat Finishing off The Maze!

The challenge ahead of rat?

For rat, there are 2 paths viz A (5 minutes) and C (3 minutes) leading to the end while paths B (8 minutes) and D (2 minutes) lead to the start again.

Since, there are 4 paths & each having equal chance of being chosen by rat, there is 1/4 th chance for each path for to be chosen by rat.

Let's assume T be the time needed for rat to finish the maze. 

But if rat selects path B or D then rat need T more time again as these paths lead to the start of the maze again.

Hence,

T = (1/4) x A + (1/4) x B + (1/4) x C + (1/4) x D

T =  (1/4) x 5 + (1/4) x (8 + T) + (1/4) x 3 + (1/4) x (2 + T)

T = (5/4) + (2) + (T/4) + (3/4) + (1/2) + (T/4)

T =  (9/2) + (T/2)

T/2 = 9/2

T = 9

That is rat needs 9 minutes to finish the maze. 

 

Row Row Row A Tiny RowBoat : Puzzle

Walter, Xavier, Yoshi, and Zeke crossed a river in a tiny rowboat. They all started on the same side of the river. They made three trips from the starting side to the destination side, and two trips from the destination side to the starting side. Here are some facts:

1. On each trip from the starting side of the river to the destination side, two people were in the boat, but only one person rowed the boat, and that person rowed the boat for the entire trip.

2. On each return trip, only one person was in the boat.

3. Walter is the weakest of the group. He could only row the boat if no one else is in it.

4. Xavier is the second weakest of the group. He can only row the boat if he is by himself or if Yoshi, the lightest of the group, is a passenger.

5. Each man rowed the boat at least once. 

Click here for SOLUTION! 

Row Row Row A Tiny RowBoat Puzzle : Solution

What was the puzzle?

We know, Walter, Xavier, Yoshi, and Zeke crossed a river in a tiny rowboat. They all started on the same side of the river. They made three trips from the starting side to the destination side, and two trips from the destination side to the starting side. 

And we have some facts:

1. On each trip from the starting side of the river to the destination side, two people were in the boat, but only one person rowed the boat, and that person rowed the boat for the entire trip.

2. On each return trip, only one person was in the boat.

3. Walter is the weakest of the group. He could only row the boat if no one else is in it.

4. Xavier is the second weakest of the group. He can only row the boat if he is by himself or if Yoshi, the lightest of the group, is a passenger.

5. Each man rowed the boat at least once. 

ANALYSIS :

1] Walter must not had rowed from start to the destination since he is weakest among as per FACT 3. Since, he had to row at least once as per FACT 5, he must had rowed a return tip.


2] The person who had rowed twice, must not had both trips from start to destination. That's because, in that case, he would have had needed third trip in form of return trip to get back to the start once again. So, he must had rowed a return trip at least once.

3] Suppose Walter is the person who rowed the boat twice. Both of his trips must be return trips as concluded in [1] above. 

So if Walter had rowed 2 return trips then each of Xavier, Yoshi and Zeke must have rowed 1 trip from start to the destination. 

If Walter had 'taken' Zeke (while Zeke rowing) across and returned then he would have had to 'take' Yoshi (while Yoshi rowing) across as Xavier being unable to row Walter as per FACT 4. And on returning back to the start after leaving Yoshi across, Walter and Xavier would have had been at the starting point. Now, Walter being unable to row with passenger & Xavier being unable to row Walter, both would have had stuck at the start point.

In short, Walter is not the person for sure who had rowed twice.

4] We know, Walter had one return trip. So the other return trip must have been rowed by someone among Xavier, Yoshi or Zeke. Moreover, the rest of three trips from start to the destination must have rowed by Xavier, Yoshi and Zeke in some order. 

That's how, the one person among Xavier, Yoshi and Zeke, must have rowed twice, one trip from source to destination and other one return trip.

5] If Xavier had rowed twice, then with one trip he must have taken Yoshi across and in other trip he must have returned as we found the fact in [2] above. So, two return trips 'occupied' by Walter and Xavier, Yoshi wouldn't have got a chance to row which is mandatory.

So, Xavier is not the person who had rowed twice.

6] Suppose Zeke is the person who had rowed twice. He must had rowed Walter across the river to give a chance to row his return trip. After Walter reaching at the start, Xavier must had rowed Yoshi across the river thereby completing his compulsory rowing trip.

Then Zeke must have returned to the start to take Walter across then Yoshi would have been the person who hadn't rowed which is against FACT 5.

Therefore, Zeke must not be the person who rowed twice.

7] So, Yoshi must be the person who rowed twice. 

POSSIBILITY 1 : Zeke took Walter across to give him row trip in return. Then, Xavier rowed Yoshi across and Yoshi returned back to take Walter across.

POSSIBILITY 2 : First Xavier rowed Yoshi across and Yoshi returned back after which Zeke takes Walter across to allow Walter to have return trip, and finally, Yoshi taking Walter across the river.

POSSIBILITY 3 : Yoshi took Walter across the river and Walter returned. Zeke rowed Walter across and Yoshi returned to give Xavier a chance to row him across.