Statement 1: In every match at least one player hit a six.
Statement 2: No two players hit sixes in the same number of matches.
Answer and solution
Correct answer: A
Solution
1. Zara hits four sixes, one in Match 1 and one in each of three other consecutive matches. Four distinct positive player totals sum to 12 and Tanya exceeds 4; the only possibility is totals 1,2,4,5. Thus Tanya has 5 and is the highest scorer.
2. Tanya scores in exactly three matches, including 4 and 8. Each even match has exactly one six, so Tanya hits 1 in 4,1 in 8 and 3 in one odd match. Zara cannot score in 4 or 8.
3. The only permitted three-match consecutive block for Zara, avoiding 4 and 8 and excluding her separate Match 1, is 5,6,7. Hence Zara scores 1 in 1,5,6,7.
4. Tanya's three-six match cannot be 7 because equal Match 3 would require too many sixes. It cannot be 3 because equality with 7 would force at least two extra sixes in 7 plus one in 2, leaving Match 3 and 7 tied for maximum. It cannot be 5 because then Match 5 would be the unique maximum, forbidden. Therefore Tanya hits 3 in Match 1.
5. Now Match 1 already has 4 sixes and Matches 4,5,6,7,8 have 1 each. Three sixes remain for Meera and Rhea. Match 2 requires 1 and equality of Matches 3 and 7 requires at least 1 in Match 3.
6. The last remaining six must be in Match 5. Putting it in Match 1 leaves Match 5=Match 3=1, forbidden; putting it in 3 or 7 would break their equality. Thus totals are 4,1,1,1,2,1,1,1. The remaining players split the three single-six slots 2,3,5 in totals 1 and 2.
7. Every match has at least one six. Tanya scores in 3 matches, Zara in 4, and Meera/Rhea in 1 and 2 matches. Both statements are true, so none is false.
G Strategy shortcut
Method: Player totals must be 1,2,4,5; place Tanya in 4/8 and Zara in 5/6/7, then use equality 3=7. Every match has at least one six. Tanya scores in 3 matches, Zara in 4, and Meera/Rhea in 1 and 2 matches. Both statements are true, so none is false.Verification: 1. Zara hits four sixes, one in Match 1 and one in each of three other consecutive matches. Four distinct positive player totals sum to 12 and Tanya exceeds 4; the only possibility is totals 1,2,4,5. Thus Tanya has 5 and is the highest scorer.
2. Tanya scores in exactly three matches, including 4 and 8. Each even match has exactly one six, so Tanya hits 1 in 4,1 in 8 and 3 in one odd match. Zara cannot score in 4 or 8.
3. The only permitted three-match consecutive block for Zara, avoiding 4 and 8 and excluding her separate Match 1, is 5,6,7. Hence Zara scores 1 in 1,5,6,7.
4. Tanya's three-six match cannot be 7 because equal Match 3 would require too many sixes. It cannot be 3 because equality with 7 would force at least two extra sixes in 7 plus one in 2, leaving Match 3 and 7 tied for maximum. It cannot be 5 because then Match 5 would be the unique maximum, forbidden. Therefore Tanya hits 3 in Match 1.
5. Now Match 1 already has 4 sixes and Matches 4,5,6,7,8 have 1 each. Three sixes remain for Meera and Rhea. Match 2 requires 1 and equality of Matches 3 and 7 requires at least 1 in Match 3.
6. The last remaining six must be in Match 5. Putting it in Match 1 leaves Match 5=Match 3=1, forbidden; putting it in 3 or 7 would break their equality. Thus totals are 4,1,1,1,2,1,1,1. The remaining players split the three single-six slots 2,3,5 in totals 1 and 2.
7. Every match has at least one six. Tanya scores in 3 matches, Zara in 4, and Meera/Rhea in 1 and 2 matches. Both statements are true, so none is false.
G Strategy: Method: Player totals must be 1,2,4,5; place Tanya in 4/8 and Zara in 5/6/7, then use equality 3=7. Every match has at least one six. Tanya scores in 3 matches, Zara in 4, and Meera/Rhea in 1 and 2 matches. Both statements are true, so none is false. Verification: 1. Zara hits four sixes, one in Match 1 and one in each of three other consecutive matches. Four distinct positive player totals sum to 12 and Tanya exceeds 4; the only possibility is totals 1,2,4,5. Thus Tanya has 5 and is the highest scorer. 2. Tanya scores in exactly three matches, including 4 and 8. Each even match has exactly one six, so Tanya hits 1 in 4,1 in 8 and 3 in one odd match. Zara cannot score in 4 or 8. 3. The only permitted three-match consecutive block for Zara, avoiding 4 and 8 and excluding her separate Match 1, is 5,6,7. Hence Zara scores 1 in 1,5,6,7. 4. Tanya's three-six match cannot be 7 because equal Match 3 would require too many sixes. It cannot be 3 because equality with 7 would force at least two extra sixes in 7 plus one in 2, leaving Match 3 and 7 tied for maximum. It cannot be 5 because then Match 5 would be the unique maximum, forbidden. Therefore Tanya hits 3 in Match 1. 5. Now Match 1 already has 4 sixes and Matches 4,5,6,7,8 have 1 each. Three sixes remain for Meera and Rhea. Match 2 requires 1 and equality of Matches 3 and 7 requires at least 1 in Match 3. 6. The last remaining six must be in Match 5. Putting it in Match 1 leaves Match 5=Match 3=1, forbidden; putting it in 3 or 7 would break their equality. Thus totals are 4,1,1,1,2,1,1,1. The remaining players split the three single-six slots 2,3,5 in totals 1 and 2. 7. Every match has at least one six. Tanya scores in 3 matches, Zara in 4, and Meera/Rhea in 1 and 2 matches. Both statements are true, so none is false.