Discussion
A book contains 10 chapters. Each chapter has a different number of pages, ranging from 20 to 30. The first chapter starts on page 1 and each chapter starts on a new page. What is the largest possible number of chapters that can begin on odd page numbers ?
Answer: B
At most 8 chapters can begin on odd pages.
Chapter 1 starts on page 1. A chapter with an even number of pages keeps the next chapter on the same parity; a chapter with an odd number of pages flips it.
The lengths are ten different numbers from 20 to 30. Only six of those are even (20, 22, 24, 26, 28, 30), so among chapters 1 to 9, whose lengths decide where chapters 2 to 10 start, at least three must have odd lengths.
Best arrangement: chapters 1 to 6 even, so chapters 1 to 7 all start on odd pages; chapter 7 odd (chapter 8 starts even), chapter 8 odd (chapter 9 back to odd), chapter 9 odd (chapter 10 starts even). That is 7 + 1 = 8 odd starts.
9 would allow at most two odd-length chapters among the first nine, which needs seven even lengths, and only six exist. 6 and 5 are possible but not the largest.
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