AMC 8 2024 Problem 7
AMC8 2024 年第 7 题
四边形与多边形★★☆☆☆
rectangle is covered without overlap by 3 shapes of tiles: , , and , shown below. What is the minimum possible number of tiles used?
一个3×7的矩形被下方3种规格分别为2×2、1×4和1×1的瓷砖所覆盖，且各瓷砖之间互不重叠。请问最少可能会用到多少个1×1的瓷砖？
- A.1
- B.2
- C.3
- D.4
- E.5
正确答案 · Correct Answer
E
解析 · Solution
解法1
我们可以排除B、C和D，因为它们不是21减去4的倍数。最后，我们看到没有办法得到A，所以解是.
设为1瓦片的数量。共有21个方格，每个2或1瓦片占据4个方格，所以，因此它要么是1要么是5。给列着色，从红色开始，然后是蓝色，交替颜色，以红色列结束。有12个红色方格和9个蓝色方格，但每个2和1形状占据相同数量的蓝色和红色方格，所以红色方格上的1瓦片必须比蓝色方格上的多3个，如果只有一个1瓦片，这是不可能的，因此答案是，这很容易验证是可行的。
假设有 种不同的 2×2 拼块， 种不同的 4×1 拼块和 种不同的 1×1 拼块。由于这些拼块的面积总和必须等于 21（整个网格的面积），我们得到 4a + 4b + c = 21. 模 4 化简得 ，即 c = 1 或 c = 5 。若 c = 1 ，则 a + b = 5 。经过一些测试，不存在有效的配对 $(a, b)$，因此答案必须是 ，且可以通过多种方式构造。T (countmath1)
由于该矩形是一个3x7的网格，其总面积（即1x1小正方形的数量）为21。为了尽可能少用1x1小正方形，我们优先使用面积较大的形状。具体来说，我们可以使用两个2x2正方形（每个覆盖4个小正方形）和两个1x4矩形（每个也覆盖4个小正方形）。这样一共填充了4+4=8个由较大形状占据的方格。填充了16个方格后，还剩下5个方格。接着我们再用5个1x1小正方形填充这些剩余方格，因此总共使用了5个1x1小正方形。所以，所需1x1小正方形的最小数量为5。
Solution 1
We can eliminate B, C, and D, because they are not 21 subtracted multiple of 4. Finally, we see that there is no way to have A, so the solution is.
Let be the number of 1 tiles. There are 21 squares and each 2 or 1 tile takes up 4 squares, so , so it is either 1 or 5. Color the columns, starting with red, then blue, and alternating colors, ending with a red column. There are 12 red squares and 9 blue squares, but each 2 and 1 shape takes up an equal number of blue and red squares, so there must be 3 more 1 tiles on red squares than on blue squares, which is impossible if there is just one, so the answer is, which can easily be confirmed to work.
Suppose there are different 2×2 tiles, different 4×1 tiles and different 1×1 tiles. Since the areas of these tiles must total up to 21 (area of the whole grid), we have 4a + 4b + c = 21. Reducing modulo 4 gives , or c = 1 or c = 5. If c = 1, then a + b = 5. After some testing, there is no valid pair $(a, b)$ that works, so the answer must be, which can be constructed in many ways. T (countmath1)
Since the rectangle is a 3x7 grid, its total area (or the number of 1x1 squares) is 21. To minimize the number of 1x1 squares used, we begin the area with larger shapes. Specifically, we can use two 2x2 squares, each covering 4 squares, and two 1x4 rectangles, each also covering 4 squares. This gives us a total of 4+4=8 4+4=8 squares filled shapes.After filling 16 squares, we have 5 squares left. We can then fill in 5 additional 1x1 squares, leaving us with a total of 5 1x1 squares. Thus, the least number of 1x1 squares required is.
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