Alok and Bhanu play the following min-max game. Given the expression N = 9 + X…
2026
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, the variable to substitute the 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.
45
- B.
50
- C.
35
- D.
20
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Correct answer: D
Work with S = X + Y − Z, so N = 9 + S.
Key idea:
Consider the last move: the final variable left will be set by Alok to maximize N. Therefore, if the last variable is X or Y he will choose 9; if it is Z he will choose 0.
Suppose Alok's first pick is a1 and Bhanu assigns it to X (the case of assigning to Y is identical). The remaining variables are Y and Z. Alok then picks a2 and Bhanu assigns it to minimize the final value. If Bhanu assigns a2 to Y the final value is 9 + a1 + a2 (because Z will be set to 0); if she assigns a2 to Z the final value is 18 + a1 − a2 (because Y will be set to 9). Bhanu chooses the smaller of these two, so Alok can guarantee max_a2 min(9 + a1 + a2, 18 + a1 − a2). Choosing a2 = 4 or 5 gives the maximum of that min, namely 13 + a1.
If Bhanu instead assigns the first pick a1 to Z, then the remaining X and Y will both be set to 9 by Alok, so Alok can force 27 − a1 (by picking a2 = 9).
Bhanu will choose the assignment that gives the smaller outcome, so the achieved value for a given a1 is min(13 + a1, 27 − a1). Alok chooses a1 to maximize this minimum.
Solve 13 + a1 = 27 − a1 → a1 = 7. At a1 = 7 both expressions equal 20, so the optimal result is S = 11 and N = 9 + 11 = 20.
Example play that achieves 20: Alok picks 7; Bhanu assigns it to X. Alok picks 5; Bhanu assigns 5 to Z (leaving Y). Alok sets Y = 9. Then N = 9 + 7 + 9 − 5 = 20.