Balancing Centrifuges with Number Theory
Balancing Centrifuges with Number Theory
复制标题
用数论平衡离心机
DOI：
10.1080/10724117.2022.2092372
复制
发表时间：
2022
期刊：
影响因子：
--
通讯作者：
Baker, Matthew H.
中科院分区：
文献类型：
--
作者：
Baker, Matthew H.
标题和图2照片由Rishav Ray提供（Reddit; https：//bit.（3zI5vvN）www. maa。org/mathhorizons数学视野|2022年9月21日www. maa。org/mathhorizons数学视野|2022年9月21日应该能够说服自己，当n= 6时，k= 1和k= 5的试管不会平衡。当n= 5或n= 7时，结果表明，只有当k= n时，才能平衡离心机。从前面的几个例子中，我们应该可以猜到答案可能与n是否是素数有关……让我们再看几个例子。当n= 8时，当且仅当k是偶数时，离心机才能平衡;当n= 9时，当k是3的倍数时，离心机才能平衡。当n= 11时，你只能在k= 11时保持平衡，但当n= 10时，有趣的事情发生了。在这种情况下，您可以找到k= 2，4，5，6，8或10的平衡离心机。例如，请注意，如果您想平衡k= 7的离心机，您可以尝试从一组平衡的五个试管开始（如图4所示，每隔一个插槽占用一个），然后在相对的孔中添加两个，但有一个问题：每对相对的孔中有一个已经被占用！我们称之为重叠问题。从某种意义上说，重叠问题是平衡离心机问题中的关键。再看一个例子。当n= 21时，当且仅当k在集合{3，6，7，9，12，14，15，18，21}中时，你可以平衡。通过考虑k= 10的情况，我们可以看到重叠问题的另一个例子。您可以尝试采用七个试管的平衡配置（如图4所示，每三个点均匀分布），然后添加三个等边三角形形状的试管，但您会发现它不起作用。
Title and figure 2 photos courtesy of Rishav Ray (Reddit; https://bit. ly/3zI5vvN) www. maa. org/mathhorizons Math Horizons| September 2022 21 www. maa. org/mathhorizons Math Horizons| September 2022 21 should be able to convince yourself that k= 1 and k= 5 test tubes won’t balance when n= 6. When n= 5 or n= 7, it turns out you can only balance the centrifuge when k= n. From these first few cases, we should guess that the answer might have something to do with whether n is prime… let’s look at a few more examples. When n= 8, you can balance the centrifuge if and only if k is even, and when n= 9, you can balance the centrifuge when k is a multiple of 3. When n= 11, you can only balance if k= 11.But something interesting happens when n= 10. In this case, you can find balanced centrifuges for k= 2, 4, 5, 6, 8, or 10. Notice, for example, that if you wanted to balance the centrifuge with k= 7, you could try to start with a balanced set of five test tubes (occupying every other slot as in figure 4) and then add two more in opposite holes, but there’sa problem: one out of each pair of opposite holes is already occupied! We will call this the overlap problem. In some sense, the overlap problem is the key subtlety in the balanced centrifuge problem. Let’s look at one more example. When n= 21, you can balance if and only if k is in the set {3, 6, 7, 9, 12, 14, 15, 18, 21}. We can see yet another illustration of the overlap problem by considering the case k= 10. You could try to take a balanced configuration of seven test tubes (evenly spaced every three spots as in figure 4) and then add three more in the shape of an equilateral triangle, but you will find that it doesn’t work.