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. 

 

Crossing The Stone Bridge : Puzzle

Three people, Ann, Ben and Jen want to cross a river from left bank to right bank. Another three people, Tim, Jim and Kim want to cross the same river from right bank to left bank.

However, there is no boat but only 1 stone bridge consisting of just 7 big stones(not tied to each other), each of which can hold only 1 person at a time. All these people have a limited jumping capacity, so that they can only jump to the stone immediately next to them if it is empty.
 
Now, all of these people are quite arrogant and so will never turn back once they have begun their journey. That is, they can only move forward in the direction of their destination. They are also quite selfish and will not help anybody traveling in the same direction as themselves.

But they are also practical and know that they will not be able to cross without helping each other. Each of them is willing to help a person coming from opposite direction so that they can get a path for their own journey ahead. With this help, a person can jump two stones at a time, such that if, say, Ann and Tim are occupying two adjacent stones and the stone next to Tim on the other side is empty, then Tim will help Ann in directly jumping to that stone, and vice versa.

Now initially the 6 people are lined up on the 7 stones from left to right as follows:

Ann Ben Jen emp Tim Jim Kim
(where emp stands for empty stone).

Your job is to find how they will cross over the stones such that they are finally lined up as follows:

Tim Jim Kim emp Ann Ben Jen

Now, find out the shortest step-wise procedure, assuming that Tim moves first.

THIS is the shortest way! 

Crossing The Stone Bridge Puzzle : Solution

What was the puzzle?

Initially the 6 people are lined up on the 7 stones from left to right as follows:
Ann Ben Jen EMP Tim Jim Kim
(where EMP stands for empty stone).

Step 1: Tim jumps to occupy the empty stone.

Ann Ben Jen Tim EMP Jim Kim

Step 2: Tim helps Jen in occupying the newly emptied stone between him and Jim. 

Ann Ben EMP Tim Jen Jim Kim
 
Step 3: Ben occupies the stone emptied by Jen.

Ann EMP Ben Tim Jen Jim Kim

Step 4: Ben helps Tim in occupying the newly emptied stone. 

Ann Tim Ben EMP Jen Jim Kim

Step 5: Jen helps Jim in occupying the empty stone. 

Ann Tim Ben Jim Jen EMP Kim

Step 6: Kim occupies the stone emptied by Jim. 

Ann Tim Ben Jim Jen Kim EMP

Step 7: Kim helps Jen in occupying the stone vacated by her. 

Ann Tim Ben Jim EMP Kim Jen
  
Step 8: Jim helps Ben in occupying the stone vacated by Jen. 

Ann Tim EMP Jim Ben Kim Jen

Step 9: Tim helps Ann in occupying the empty stone.

EMP Tim Ann Jim Ben Kim Jen

Step 10: Tim jumps to the stone emptied by Ann.

Tim EMP Ann Jim Ben Kim Jen

Step 11: Ann helps Jim in occupying the stone vacated by Tim.

Tim Jim Ann EMP Ben Kim Jen

Step 12: Ben helps Kim in occupying the stone vacated by Jim. 

Tim Jim Ann Kim Ben EMP Jen

Step 13: Ben occupies the empty stone.

Tim Jim Ann Kim EMP Ben Jen

Step 14: Kim helps Ann in occupying the stone emptied by Ben. 

Tim Jim EMP Kim Ann Ben Jen

Step 15: Kim jumps to the stone emptied by Ann.

Tim Jim Kim EMP Ann Ben Jen

This is exactly what we wanted!

 

A Determined Cat on a Ladder!

A ladder is leaning against a wall. On the center rung is a cat. She must be a very determined cat, because she remains on that rung as we draw the foot of the ladder away from the tree until the ladder is lying flat on the ground. What path does the cat describe as she undergoes this indignity?

She follows this path!

A Path Followed by Determined Cat

What was the question?

 Interestingly, the cat follows the circular path whose center is at the foot of tree. 

Actually, as ladder is drawn out a series of right triangles with the same hypotenuse (the ladder) are created with respect to the foot of tree.

The point of hypotenuses where cat is sitting determinedly will be always at the same distance from all 3 vertices. So if all such point are joined then we get a circular path having center at the foot of tree. 

(Figure is for illustration purpose only & may not have accurate measurements.