TIFR CSE 2019 | Part A | Question: 10

Question

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:

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

• A. Avni has a winning strategy
• B. Badal has a winning strategy
• C. Both of them have a winning strategy
• D. Neither of them has a winning strategy
• E. The Player that picks $9$ has a winning strategy

Comments

I think this question is similar to playing tic-tac-toe. IF so, the answer would be D.

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OneZero yes same $D$

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if such type question comes in GATE :)

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https://puzzling.stackexchange.com/questions/48132/finding-digits-that-sum-to-15 

Explained here 

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@Mk Utkarsh

why not C)?

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

check the explanation provided by Sayan Bose

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

Answer 1

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.

More read: https://puzzling.stackexchange.com/questions/48132/finding-digits-that-sum-to-15