N=x-y-z

Where x,y,z are variable representation of some digit (0 to 9) Alok is like to maximize N while Bhanu Is like to minimize it towards this end ,Alok chooses a single digit number and bhanu substitute this for a variable of her choice (x,y or z)also chooses the next value and Bhanu , the variable to substitute the value .finally ALOK proposes the value for the remaining variable .assuming both playing to their optimal strategies,the value of N at the end of the game would be

To determine the value of N at the end of the game, we need to consider the optimal strategies of both Alok and Bhanu.

Let's analyze the game step by step:

Step 1:
Alok chooses a single-digit number. Since Alok wants to maximize N, he will choose the largest possible digit, which is 9.

Step 2:
Bhanu now has to substitute this value for one of the variables (x, y, or z). Bhanu wants to minimize N, so she will substitute 9 for the variable that will result in the smallest value of N.

To minimize N, Bhanu should substitute 9 for the variable that appears with the largest coefficient in the expression. Since we don't have information about the coefficients, we cannot determine which variable to substitute. So, let's assume Bhanu substitutes 9 for the variable x.

Now we have N = 9 - y - z.

Step 3:
It is now Alok's turn to choose the next value. Alok wants to maximize N, so he will choose the largest possible digit, which is again 9.

Step 4:
Bhanu needs to substitute this value for one of the remaining variables (y or z). Bhanu wants to minimize N, so she will substitute 9 for the variable that will result in the smallest value of N.

Again, Bhanu should analyze the coefficients to determine which variable to substitute. Since we don't have coefficient information, let's assume Bhanu substitutes 9 for the variable y.

Now we have N = 9 - 9 - z.

Step 5:
Alok proposes the value for the remaining variable. Since Alok wants to maximize N, he will choose the largest possible digit, which is 9.

Substituting this value, we have N = 9 - 9 - 9.

Simplifying the expression, we get N = -9.

Therefore, at the end of the game, the value of N would be -9.