# (Puzzles) Challenging Mathematical Puzzles (27)

**Puzzles : Challenging Mathematical Puzzles**

**82. In the town called Alibaug, the following facts are true:
* No two inhabitants have exactly the same number of hairs.
* No inhabitants has exactly 2025 hairs.
* There are more inhabitants than there are hairs on the head of any one inhabitants.
What is the largest possible number of the inhabitants of Alibaug?
Answer : - **2025

It is given that no inhabitants have exactly 2025 hairs. Hence there are 2025 inhabitants with 0 to 2024 hairs in the head.

Suppose there are more than 2025 inhabitants. But these will violate the condition that "There are more inhabitants than there are hairs on the head of any one inhabitants." As for any number more than 2025, there will be same number of inhabitants as the maximum number of hairs on the head of any inhabitant.

**83. There are four groups of Mangoes, Apples and Bananas
as follows:
Group I : 1 Mango, 1 Apples and 1 Banana
Group II : 1 Mango, 5 Apples and 7 Bananas
Group III : 1 Mango, 7 Apples and 10 Bananas
Group IV : 9 Mango, 23 Apples and 30 Bananas
Group II costs Rs 300 and Group III costs Rs 390.
Can you tell how much does Group I and Group IV cost?**

**Answer:-**Group I costs Rs 120 and Group IV costs Rs 1710

Assume that the values of one mango, one apple and one banana are M, A and B respectively.

From Group II : M + 5A + 7B = 300

From Group III : M + 7A + 10B = 390

Subtracting above to equations : 2A + 3B = 90

For Group I :

= M + A + B

= (M + 5A + 7B) - (4A + 6B)

= (M + 5A + 7B) - 2(2A + 3B)

= 300 - 2(90)

= 300 - 180

= 120

Similarly, for Group IV :

= 9M + 23A + 30B

= 9(M + 5A + 7B) - (22A + 33B)

= 9(M + 5A + 7B) - 11(2A + 3B)

= 9(300) - 11(90)

= 2700 - 990

= 1710

Thus, Group I costs Rs 120 and Group IV costs Rs 1710

**84. Tic-Tac-Toe is being played. One 'X' has been placed in one of the corners. No 'O' has been placed yet.
Where does the player that is playing 'O' has to put his first 'O' so that 'X' doesn't win?
Assume that both players are very intelligent. Explain your answer
Answer : -**

"O" should be placed in the center.

Let's number the positions as:

1 | 2 | 3

---------

4 | 5 | 6

---------

7 | 8 | 9

It is given that "X" is placed in one of the corner position. Let's assume that its at position 1.

Now, let's take each position one by one.

* If "O" is placed in position 2, "X" can always win by choosing position 4, 5 or 7.

* If "O" is placed in position 3, "X" can always win by choosing position 4, 7 or 9.

* If "O" is placed in position 4, "X" can always win by choosing position 2, 3 or 5.

* If "O" is placed in position 6, "X" can always win by choosing position 3, 5 or 7.

* If "O" is placed in position 7, "X" can always win by choosing position 2, 3 or 9.

* If "O" is placed in position 8, "X" can always win by choosing position 3, 5 or 7.

* If "O" is placed in position 9, "X" can always win by choosing position 3, or 7.

If "O" is placed in position 5 i.e. center position, "X" can't win unless "O" does something foolish ;))

Hence, "O" should be placed in the center.