Alok and Bhanu play the following min-max game. Given the expression N = 9 + X…
2024
Alok and Bhanu play the following min-max game. Given the expression N = 9 + X + Y – Z. Where X, Y and Z are variables representing single digits (0 to 9), Alok would like to maximize N while Bhanu would like to minimize it. Towards this end, Alok chooses a single digit number and Bhanu substitutes this for a variable of her choice (X, Y or Z). Alok then chooses the next value and Bhanu put, the variable to substitute the next value. Finally Alok proposes the value for the remaining variable. Assuming both play to their optimal strategies, the value of N at the end of the game would be?
- A.
0
- B.
27
- C.
18
- D.
20
Attempted by 20 students.
Show answer & explanation
Correct answer: C
Answer: 18
Reasoning:
When Alok picks a digit, Bhanu chooses which variable (X, Y or Z) receives it. To minimize N = 9 + X + Y - Z, Bhanu will always assign the largest of the three chosen digits to Z.
If the three chosen digits are a, b, c then after Bhanu's placements the value is
N = 9 + X + Y - Z = 9 + (a + b + c) - 2*max(a, b, c).
To maximize this expression Alok should maximize the sum a + b + c while keeping the largest term as small as possible. The best choice is a = b = c = 9.
With 9, 9, 9 we get N = 9 + (27) - 2*9 = 9 + 27 - 18 = 18. No other combination of three digits gives a larger value.
Example of play that achieves 18: Alok picks 9 (Bhanu puts it at Z), Alok picks 9 (Bhanu puts it at X or Y), Alok picks 9 for the remaining variable. Final X = 9, Y = 9, Z = 9 so N = 18.