We can cluster in one dimension as well as in many dimensions. In this problem, we are going to cluster numbers on the real line. The particular numbers (data points) are 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100, i. e., the squares of 1 through 10. We shall use a k-means algorithm, with two clusters. You can verify easily that no matter which two points we choose as the initial centroids, some prefix of the sequence of squares will go into the cluster of the smaller and the remaining suffix goes into the other cluster. As a result, there are only nine different clusterings that can be achieved, ranging from {1}{4,9,...,100} through {1,4,...,81}{100}. We then go through a reclustering phase, where the centroids of the two clusters are recalculated and all points are reassigned to the nearer of the two new centroids. For each of the nine possible clusterings, calculate how many points are reclassified during the reclustering phase. Identify in the list below the pair of initial centroids that results in exactly one point being reclassified.
a) 36 and 100
b) 1 and 16
c) 1 and 25
d) 9 and 81