D.R.KAPRELAR DISCOVERIES

Kaprekar constant

In 1949, Kaprekar discovered an interesting property of the number 6174, which was subsequently named the Kaprekar constant. He showed that 6174 is reached in the limit as one repeatedly subtracts the highest and lowest numbers that can be constructed from a set of four digits that are not all identical. Thus, starting with 1234, we have:

4321 − 1234 = 3087, then

8730 − 0378 = 8352, and

8532 − 2358 = 6174.

Repeating from this point onward leaves the same number (7641 − 1467 = 6174). In general, when the operation converges it does so in at most seven iterations.

A similar constant for 3 digits is 495. However, in base 10 a single such constant only exists for numbers of 3 or 4 digits; for other digit lengths or bases other than 10, the Kaprekar's routine algorithm described above may in general terminate in multiple different constants or repeated cycles, depending on the starting value.

Kaprekar number

Another class of numbers Kaprekar described are the Kaprekar numbers. A Kaprekar number is a positive integer with the property that if it is squared, then its representation can be partitioned into two positive integer parts whose sum is equal to the original number (e.g. 45, since 45²=2025, and 20+25=45, also 9, 55, 99 etc.) However, note the restriction that the two numbers are positive; for example, 100 is not a Kaprekar number even though 100²=10000, and 100+00 = 100. This operation, of taking the rightmost digits of a square, and adding it to the integer formed by the leftmost digits, is known as the Kaprekar operation.

Some examples of Kaprekar numbers in base 10, besides the numbers 9, 99, 999, …, are (sequence A006886 in the OEIS):

Number	Square	Decomposition
703	703² = 494209	494+209 = 703
2728	2728² = 7441984	744+1984 = 2728
5292	5292² = 28005264	28+005264 = 5292

       P.C. MAHALANOBIS CONTRIBUTIONS

Mahalanobis Distance

                         Mahalanobis Distance is one of the most widely used metric to find how much a point diverges from a distribution, based on measurements in multiple dimensions. It is widely used in the field of cluster analysis and classification. It was first proposed by Mahalanobis in 1930 in context of his study on racial likeness. From a chance meeting with Nelson Annandale, then the director of the Zoological Survey of India, at the 1920 Nagpur session of the Indian Science Congress led to Annandale asking him to analyse anthropometric measurements of Anglo-Indians in Calcutta. Mahalanobis had been influenced by the anthropometric studies published in the journal Biometrika and he chose to ask the questions on what factors influence the formation of European and Indian marriages. He wanted to examine if the Indian side came from any specific castes. He used the data collected by Annandale and the caste-specific measurements made by Herbert Risley to come up with the conclusion that the sample represented a mix of Europeans mainly with people from Bengal and Punjab but not with those from the Northwest Frontier Provinces or from Chhota Nagpur. He also concluded that the intermixture more frequently involved the higher castes than the lower ones. This analysis was described by his first scientific paper in 1922. During the course of these studies he found a way of comparing and grouping populations using a multivariate distance measure. This measure, denoted "D²" and now eponymously named Mahalanobis distance, is independent of measurement scale. Mahalanobis also took an interest in physical anthropology and in the accurate measurement of skull measurements for which he developed an instrument that he called the "profiloscope".