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Avni and Badal alternately choose numbers from the set $\{1,2,3,4,5,6,7,8,9\}$  without replacement (starting with Avni). The first person to choose numbers of which any $3$ sum to $15$ wins the game (for example, Avni wins if she chooses the numbers $8, 3, 5, 2$ since $8+5+2=15$). A player is said to have a winning strategy if the player can always win the game, no matter what the other player does. Which of the following statements is TRUE?

As a hint, there are exactly $8$ ways in which $3$ numbers from the set $\{1,2,3,4,5,6,7,8,9\} $ can sum up to $15$, shown as the three rows, the three columns, and the two diagonals in the following square:

$$\begin{array} & 8 & 1 & 6  \\  3 & 5 & 7 \\  4 & 9 & 2 \end{array}$$

  1. Avni has a winning strategy
  2. Badal has a winning strategy
  3. Both of them have a winning strategy
  4. Neither of them has a winning strategy
  5. The Player that picks $9$ has a winning strategy
in Numerical Ability by
edited by | 361 views
I think this question is similar to playing tic-tac-toe. IF so, the answer would be D.

 yes same $D$

if such type question comes in GATE :)

@Mk Utkarsh

why not C)?


there exist no strategy where you can win tic tac toe everytime

check the explanation provided by 

Yes there is no guaranteed winning strategy. There is a strategy to avoid defeat though

1 Answer

+3 votes
  1. $9+5+1$
  2. $9+4+2$
  3. $8+6+1$
  4. $8+5+2$
  5. $8+4+3$
  6. $7+6+2$
  7. $7+5+3$
  8. $6+5+4$

Consider each digits from $1$ to $9$

  1. For $1$, $\{5,9\}$ and $\{8,6\}$ are the winning combinations. So, if a player picks $1$ the other player should avoid picking any number from $\{5,6,8,9\}$ and if the first player then picks $5$ the second player must pick $9$ and if the first player picks $8$ the second player must pick $6$ (same for reverse order too) thus blocking all the winning combinations.
  2. Like above for all digits from $2$ to $9$ the winning combinations can be blocked by the other player because if one element is picked there are no common elements in the winning combinations.

So, option D is correct. Neither has a winning strategy.

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