Question

Five horses, Red, White, Grey, Black and Spotted participated in a race. As per the rules of the race, the persons betting on the winning horse get four times the bet amount and those betting on the horse that came in second get thrice the bet amount. Moreover, the bet amount is returned to those betting on the horse that came in third, and the rest lose the bet amount. Raju bets Rs. 3000, Rs. 2000 and Rs. 1000 on Red, White and Black horses respectively and ends up with no profit and no loss. Which of the following cannot be true?

• A. At least two horses finished before Spotted
• B. Red finished last
• C. There were three horses between Black and Spotted
• D. There were three horses between White and Red
• E. Grey came in second

Hint:

Raju stakes Rs. 6000 in all and gets back exactly Rs. 6000. Work out which finishing positions of Red, White and Black make the payouts add up to 6000, then test each option.

Answer 1

The Correct Option is D

Step 1: Set up the payout equation:
Work in thousands of rupees. Raju bets \(3\) on Red, \(2\) on White and \(1\) on Black, so he stakes \(6\) in all. No profit and no loss means he gets back exactly \(6\). A horse that wins pays \(4 \times\) its bet, second pays \(3 \times\), third pays back \(1 \times\) the bet, and 4th or 5th pays \(0\).

Step 2: Find the possible results for Red:
Red first would give \(12\) and Red second would give \(9\), both more than \(6\), so Red is third (returns \(3\)) or 4th/5th (returns \(0\)).

Step 3: List all cases:
Case A, Red third (returns \(3\)): White and Black must return \(3\) in all. White first gives \(8\) and White second gives \(6\), both too much, and third place is taken by Red, so White is 4th/5th and returns \(0\). Then Black must return \(3\), that is, Black is second. So: Red 3rd, Black 2nd, White 4th or 5th.
Case B, Red 4th or 5th (returns \(0\)): White and Black must return \(6\). White first gives \(8\), too much. White second gives \(6\), so Black returns \(0\): White 2nd, Red and Black in the last two places. White third gives \(2\), so Black must return \(4\): Black first. So: Black 1st, White 3rd, Red 4th/5th.

Step 4: What every case has in common:
In all cases neither Red nor White finishes first. "Three horses between White and Red" means they are in positions 1 and 5, so one of them would have to be first. This is impossible.

Step 5: Check that the other options can be true:
Take the order Black 1st, Grey 2nd, White 3rd, Spotted 4th, Red 5th. It satisfies (1) (at least two horses before Spotted), (2) (Red last) and (5) (Grey second). The order Black 1st, Grey 2nd, White 3rd, Red 4th, Spotted 5th has three horses between Black and Spotted, so (3) is possible. All of these orders fit Case B.

Final Answer:
The correct option is (4): There were three horses between White and Red.