HMMT 十一月 2017 · 冲刺赛 · 第 19 题
HMMT November 2017 — Guts Round — Problem 19
题目详情
- [ 9 ] The skeletal structure of coronene, a hydrocarbon with the chemical formula C H , is shown 24 12 below. Each line segment between two atoms is at least a single bond. However, since each carbon (C) requires exactly four bonds connected to it and each hydrogen (H) requires exactly one bond, some of the line segments are actually double bonds. How many arrangements of single/double bonds are there such that the above requirements are satisfied? (Rotations and reflections of the same arrangement are considered distinct.)
解析
- [ 9 ] The skeletal structure of coronene, a hydrocarbon with the chemical formula C H , is shown 24 12 below. Each line segment between two atoms is at least a single bond. However, since each carbon (C) requires exactly four bonds connected to it and each hydrogen (H) requires exactly one bond, some of the line segments are actually double bonds. How many arrangements of single/double bonds are there such that the above requirements are satisfied? Proposed by: Yuan Yao Answer: 20 Note that each carbon needs exactly one double bond. Label the six carbons in the center 1 , 2 , 3 , 4 , 5 , 6 clockwise. We consider how these six carbons are double-bonded. If a carbon in the center is not double-bonded to another carbon in the center, it must double-bond to the corresponding carbon on the outer ring. This will result in the outer ring broken up into (some number of) strings instead of a loop, which means that there will be at most one way to pair off the outer carbons through double-bonds. (In fact, as we will demonstrate later, there will be exactly one way.) Now we consider how many double bonds are on the center ring. • 3 bonds. There are 2 ways to pair of the six carbons, and 2 ways to pair of the outer ring as well, for 4 ways in total. • 2 bonds. Then either two adjacent carbons (6 ways) or two diametrically opposite carbons (3 ways) are not double-bonded, and in the former case the outer ring will be broken up into two “strands” with 2 and 14 carbons each, while in the latter case it will be broken up into two strands both with 8 carbons each, and each produce one valid way of double-bonding, for 9 ways in total. • 1 bond. There are 6 ways to choose the two double-bonded center carbon, and the outer ring will be broken up into four strands with 2, 2, 2, 8 carbons each, which gives one valid way of double-bonding, for 6 ways in total. • 0 bonds. Then the outer ring is broken up into six strands of 2 carbons each, giving 1 way. Therefore, the number of possible arrangements is 4 + 9 + 6 + 1 = 20. Note: each arrangement of single/double bonds is also called a resonance structure of coronene.