AMC 8 2013 Problem 22
AMC8 2013 年第 22 题
基本计数原理★★★☆☆
Toothpicks are used to make a grid that is 60 toothpicks long and 32 toothpicks wide. How many toothpicks are used altogether?
下图所示的网格是用牙签做成的，网格长为 60 根牙签，宽为 32 根牙签。则总共用了多少根牙签？
- A.1920
- B.1952
- C.1980
- D.2013
- E.3932
正确答案 · Correct Answer
E
解析 · Solution
Solution 1
There are 61 vertical columns with a length of 32 toothpicks, and there are 33 horizontal rows with a length of 60 toothpicks, because 32 and 60 are the number of intervals. You can verify this by trying a smaller case, i.e. a 3×4 grid of toothpicks, with 3×3 and 2×4. Thus, our answer is . by Theraccoon: The person who posted this answer did not include their name. Minor edit by - Common Sense With a quick mental calculation, 60 * 30 yields 1800, which is roughly where 4 of our 5 answer choices lie in. However, we can tell that each square would require at most 2 toothpicks that uniquely belong to itself, so the answer would be which would be roughly 3932. -pearlseal34, Minor Edit by MagentaButterfly29 - Deriving a Formula If you wanted to find the number of toothpicks in a x by y rectangle you will notice a pattern. We can separate the toothpicks into 2 groups, vertical and horizontal toothpicks and add them together. There is x(y+1) vertical toothpick that are vertical and y(x+1) horizontal toothpicks.If my explantation doesn't make sense try drawing it out. You can add these two for the total amount of toothpicks. Now we can plug in 60 into x and 32 into y and solve. 60(32+1)+32(60+1)= 60(32)+60+32(60)+32=3932 which is 3932. -ZenJacob - Border vs. Internal toothpicks The total area of the 60 X 32 grid is 1920 square units. If each square was separated, we would need 1920 * 4 = 7680 toothpicks. However, once all the squares are joined to form a grid, each internal toothpick forms the edge of two adjacent squares. The perimeter/border of the grid contains 60 + 60 + 32 + 32 = 184 sticks. So, 7680 - 184 = 7496 toothpicks are internal toothpicks. Since each internal toothpick forms the edge of two adjacent squares, we need only 7496 / 2 = 3748 of them. So, the total toothpicks needed are 3748 (internal) + 184 (border) which is 3932. -AbbasK
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