**Why this master plan needed?**

**99**can be guaranteed to save! How?

Even if the person behind calls out the color of the hat that next person is wearing

**both**would be survived only if they are wearing same color of hat.

So how 99 can be saved?

For a simplicity, let's assume there are only

**10**wise men & (only) assume we are among them. Now, we need to make a master plan to survive from this game of death.

One of us need to agree to sacrifice his life to save 9 of us & this person would be the first one in line. He will be survived of he has good luck.

The

**first person**in line should shout

**RED**if he founds number of

**RED**hats

**even**otherwise he should shout

**BLUE.**Now if he has

**good luck**then the hat color of his own hat would match & he would be survived.

The clue given by the

**first person**is very important. Right from

**second person**everyone need to count number of

**RED**hats

**in fro**

**nt**of him. Additionally, the next person need to keep track of number of

**RED**hats that people

**behin**

**d**him are wearing.

In other words, every person need to think that how the first person counted number of

**RED**hats. Let you be that person. Now the first person count number of

**RED**hats behind you + your hat color (if it's

**RED**) + number of

**RED**hats in front of you.

Now,in our example, the first person counts number of

**RED**hats as 4 & says

**RED**

**.**Luckily his own hat color too is

**RED**& hence he would be survived.Every other one

**should**

**not**count the color of hat that first person is wearing.

Now the

**second person**counts number of

**RED**hats again as 4. Now he should think that if he had

**RED**hat as well then the first person would have counted 5

**RED**hats. So he need to deduce that his hat must be

**BLUE.**

The king would move to the next person.

The

**third person**need to keep in mind that what second said. Now, here second said

**BLUE**& the number of

**RED**in front of third person is 3 i.e.

**odd.**Hence, he need to deduce that his hat color is

**RED**as only in that case the number of

**RED**hats the first person saw would be

**even**i.e.4. Otherwise, the first person would have counted only 3

**RED**hats if third person had

**BLUE**hat.

Now the

**fourth person**keeps the track of number of

**RED**he had heard previously (here only once). Now he counts number of

**RED**hats in front of him (here 3) then add it to the number of

**RED**s he had heard (here it is 1). So total number of

**R**

**ED**hats according to him is 4 (excluding first person's hat). That is

**e**

**ven**as first person pointed. Hence he must deduce that he has

**BLUE**hat. If he had

**RED**one then first person would have counted 5 number of

**RED**hats i.e. odd number & would have called out

**BLUE.**

In this way, fifth & sixth need to guess correctly the color of their own hat.

Let's take look how

**se**

**venth person**would respond as per out plan. He can see only 1

**R**

**ED**hat in front of him. Additionally, he had heard 3

**RED**s (from 3,4,6). Adding those, according to him number of

**RED**hats (excluding first person's hat) is 3 + 1 = 4. Again, it's

**even**similar to that counted by first person.

In short, if the person has data of number of

**RED**hats (from behind added to that in front) as

**even**(similarly as pointed by first person) then his hat color must be

**BLUE**otherwise

**RED.**

On the similar note, if the

**first person**calls out

**BLUE**& if the person having number of

**RED**hats (adding number of

**RED**hats from behind to that of in front)

**odd**(similarly as pointed by first person) then his hat must be

**BLUE**

**.**And if the person finds that number

**e**

**ven**then he must be wearing

**BLUE**hat.

One thing to note here is that whatever happened with the previous person

**d**

**oesn't**matter here. In fact, every one would be

**abs**

**olutely sure**with the survival of every other person except the first person.

What the most important is

**first person's call**& every person's

**calc**

**ulations**

**.**The later must be very accurate to ensure life of all.

We can,

**'encode'**the

**e**

**ven or odd**nature of number of

**RED**hats or

**BLUE**hats which the first person is going to indicate and accordingly deduce method to correctly determine the color of hat that each person is wearing.

This is how we can ensure

**9**survived in

**10**

**.**With the same plan,

**99**can be guaranteed to be survived in

**100.**

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