Arrange Positions Around The Round-Table

King Arthur and his eleven honorable knights must sit on a round-table. In how many ways can you arrange the group, if no honorable knight can sit between two older honorable knights?

Here are the possible combinations!

Source 

Possible Positions Around The Round Table

What was the challenge?

If king K is sitting at the center then the youngest knight must sit to right or left of the king i.e. 2 possible positions for him.

The second youngest knight now can sit either left or right of the group of 2 made above.

The third youngest knight now can sit either left or right of the group of 3 made above.

And so on.

That is every knight has 2 possible positions except the oldest knight who will have only 1 position left.

This will be make sure each of the knight (except youngest one) has at least 1 younger neighbor (youngest one has king as one neighbor).

So after putting youngest in 2 possible ways the next youngest can be put another 2 possible ways. That is 4 possible combinations for 2.

Similarly, for arranging 3 knights' positions there can be 2^3 = 8 possible combinations.

This way for 10 knights (excluding the oldest which will have only 1 seat available at the end) there are 2^10 = 1024 possible combinations.

Sharing The Driving Time!

John and Mary drive from Westville to Eastville. John drives the first 40 miles, and Mary drives the rest of the way. That afternoon they return by the same route, with John driving the first leg and Mary driving the last 50 miles.

Who drives the farthest, and by what distance?

'This' is the person who drives the farthest!


Source

Uneven Sharing of Driving Time!

But actual how it was shared?

Let's assume for a moment, the distance between Westville and Eastville is 50 miles.

In this case, John drives only 40 miles while Mary drives 10 + 50 = 60 miles. For any distance beyond 50 miles, Mary drives east equal distance as John drives west.

So in any case the difference will remain same of 20 miles. So Mary drives the farthest by 20 miles.

What is Color of His Hat?

There is a basket full of hats. 3 of them are white and 2 of them are black. There are 3 men Tom, Tim, and Jim. They each take a hat out of the basket and put it on their heads without seeing the hat they selected or the hats the other men selected.

The men arrange themselves so Tom can see Tim and Jim’s hats, Tim can see Jim’s hat, and Jim can’t see anyone’s hat.

Tom is asked what color his hat is and he says he doesn’t know.

Tim is asked the same question, and he also doesn’t know.

Finally, Jim is asked the question, and he does know.

What color is his hat?

Know color of his hat! 

'THIS' is The Color of His Hat!

What was the challenge?

Since there are only 2 black hats if Tom had saw 2 black hats (on heads of Tim & Jim) then he would have realized that he must be wearing white hat. Since, he says he doesn't know color of his hat that mean he can see either 2 white  hats or 1 black & 1 white hat on heads of other 2.

Now when Tim is saying he doesn't know color of his hat that means Jim must not be wearing black hat. If Jim had black hat then Tim would know that his color of hat is white but can't be black again as in that case Tom would have identified color of his hat in first attempt.

Hence, Jim must be wearing white hat! 

Plan an Unbeatable Strategy

Two people play a game of NIM. There are 100 matches on a table, and the players take turns picking 1 to 5 sticks at a time. The person who takes the last stick wins the game. (Both players has to make sure that the winner would be picking only 1 stick at the end) 

Who has a winning strategy?

And what must be winning strategy in the person who takes the last stick looses?

This could be the winning strategy!