Three players named Arun, Babita and Chandu play a game by tossing an unbiased coin in turn whose rules are as follows: Initially each player has 100 candies. If the tossing of this coin results in a 'head' then the player who tosses receives 10 candies from each of the other two players, whereas, if the toss results in a 'tail' then the player who tosses has to give away 20 candies to each of the other two players. A player with the highest number of candies at the end will be the winner. The game is started by Arun, followed by Babita and finally stopped after Chandu's turn. Given that Arun is the only winner, which of the following hold(s) true?
Game Rules and Setup
Initial state: Arun (A), Babita (B), and Chandu (C) each start with 100 candies.
The game proceeds in turns: Arun, then Babita, then Chandu. The game stops after Chandu's turn.
Candy transfer rules:
- Head (H): The player tossing receives 10 candies from each of the other two players. Net change for tosser: $+20$. Net change for others: $-10$ each.
- Tail (T): The player tossing gives 20 candies to each of the other two players. Net change for tosser: $-40$. Net change for others: $+20$ each.
The winner is the player with the most candies at the end. We are given that Arun is the *only* winner ($A_f > B_f$ and $A_f > C_f$).
Analyzing Candy Transfers
Let $A_0, B_0, C_0$ represent the initial candy counts (100 each).
Let $t_A, t_B, t_C$ denote the outcome (H or T) of the tosses by Arun, Babita, and Chandu, respectively.
The final candy count for each player ($A_f, B_f, C_f$) is their initial count plus the net change from their own toss and the changes caused by the other players' tosses.
Net change calculation:
- Arun's net change ($\Delta A$): Change from his toss + Change from Babita's toss + Change from Chandu's toss.
- Babita's net change ($\Delta B$): Change from Arun's toss + Change from her toss + Change from Chandu's toss.
- Chandu's net change ($\Delta C$): Change from Arun's toss + Change from Babita's toss + Change from his toss.
Identifying the Winning Scenario
We need to find the outcome $(t_A, t_B, t_C)$ where Arun is the sole winner. Let's analyze the possible outcomes:
| Outcome $(t_A, t_B, t_C)$ | Net Change for Arun ($\Delta A$) | Net Change for Babita ($\Delta B$) | Net Change for Chandu ($\Delta C$) | Final Candies $(A_f, B_f, C_f)$ | Winner(s) |
| (H, H, H) | $20 - 10 - 10 = 0$ | $-10 + 20 - 10 = 0$ | $-10 - 10 + 20 = 0$ | $(100, 100, 100)$ | None |
| (H, H, T) | $20 - 10 + 20 = 30$ | $-10 + 20 + 20 = 30$ | $-10 - 10 - 40 = -60$ | $(130, 130, 40)$ | Arun, Babita |
| (H, T, H) | $20 + 20 - 10 = 30$ | $-10 - 40 - 10 = -60$ | $-10 + 20 + 20 = 30$ | $(130, 40, 130)$ | Arun, Chandu |
| (H, T, T) | $20 + 20 + 20 = 60$ | $-10 - 40 + 20 = -30$ | $-10 + 20 - 40 = -30$ | $(160, 70, 70)$ | Arun |
| (T, H, H) | $-40 - 10 - 10 = -60$ | $+20 + 20 - 10 = 30$ | $+20 - 10 + 20 = 30$ | $(40, 130, 130)$ | Babita, Chandu |
| (T, H, T) | $-40 - 10 + 20 = -30$ | $+20 + 20 + 20 = 60$ | $+20 - 10 - 40 = -30$ | $(70, 160, 70)$ | Babita |
| (T, T, H) | $-40 + 20 - 10 = -30$ | $+20 - 40 - 10 = -30$ | $+20 + 20 + 20 = 60$ | $(70, 70, 160)$ | Chandu |
| (T, T, T) | $-40 + 20 + 20 = 0$ | $+20 - 40 + 20 = 0$ | $+20 + 20 - 40 = 0$ | $(100, 100, 100)$ | None |
The only scenario where Arun is the sole winner is when the sequence of tosses is (Head, Tail, Tail). In this case, the final candy counts are: Arun $A_f = 160$, Babita $B_f = 70$, Chandu $C_f = 70$.
Evaluating the Options
Let's check the given options using the final counts $A_f = 160, B_f = 70, C_f = 70$:
- Arun has 90 candies more than Chandu: Is $A_f = C_f + 90$? $160 = 70 + 90$. This statement is true.
- Both Babita and Chandu have the same number of candies: Is $B_f = C_f$? $70 = 70$. This statement is true.
- Arun has 90 candies more than Babita: Is $A_f = B_f + 90$? $160 = 70 + 90$. This statement is true.
- Babita and Chandu together have 120 candies: Is $B_f + C_f = 120$? $70 + 70 = 140$. This statement is false.
The options that hold true are 1, 2, and 3.
Important Questions from Numerical Reasoning
- $P, Q, R, S, X$, and $Y$ are distinct single-digit whole numbers taking values from 0 to 9.
$PQ$ is a two-digit number with $Q$ being in the units place and $P$ in the tens place. Similarly, $RS$ is a two-digit number.
It is known that $PQ$ and $RS$ are consecutive numbers and
$(PQ)^2 + (RS)^2 = XYP$, with $XYP$ being a three-digit number.
The value of $Y$ is __________ - Let $p_1$ and $p_2$ denote two arbitrary prime numbers. Which one of the following statements is correct for all values of $p_1$ and $p_2$?
If $\oplus \div \odot = 2$, $\oplus \div \triangle = 3$, $\odot + \triangle = 5$, and $\Delta \times \otimes = 10$,
then the value of $(\otimes - \oplus)^2$ is:- The remainder when $98!$ is divided by $101$ is equal to ________
- A 'frabjous' number is defined as a 3 digit number with all digits odd, and no two adjacent digits being the same. For example, 137 is a frabjous number, while 133 is not. How many such frabjous numbers exist?
