# The number 200 | Science

This is number 200 of the articles of *The game of science*, which is a good excuse to deal with this doubly round number. To begin, we can apply it to the riddle of the ugly, stupid and bad men of last week: if there is 70% of each, we have 210 "qualities" to distribute among 100 men, and the most homogeneous distribution begins by attributing two to each one, with which we will have distributed 200 and we will have 10 more; therefore, there will be at least 10 men out of 100 - that is, 10% - who will possess all three characteristics at the same time.

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Advertise HereRegarding the coincidence of age and number of hairs, and since we are commemorating a double hundred, we round up the data in the hundreds and consider that in a human head there is a maximum of 100,000 hairs, which in Spain there are fifty hundred hairs. millions of people and who live a maximum of 100 years. There will therefore be 100 x 100,000 = 10,000,000 possible age-hair pairings. If all the possibilities were distributed homogeneously, we would have five people in each group, and therefore there is a minimum of five people of the same age and with the same number of hair. In practice, there will be very little nourished groups, such as 100-100,000 (centenarians with 100,000 hairs) and others very numerous, such as 50-0 (fifty-bald bald).

In any group of people, there are at least two who have slept with the same number of group members. In effect, if we call n the number of people in the group, the maximum possible pairings for each is n - 1 and the minimum 0; there are, therefore, n possibilities, so at first glance it seems that each person could have a different possibility: but possibility 0 and n - 1 are mutually exclusive: if someone has not slept with anyone, nobody can have lying with everyone.

### An excessive number

The 200 is an "abundant" or "excessive" number, which means that the sum of its divisors (including 1 and itself) is greater than twice the number itself; indeed, the divisors of 200 are 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, and 200, and their sum is 465, which is May than 400.

And, to finish, the usual themed riddles:

Can you convert 200 to a prime number by changing only one of its numbers? And in the event that this is not possible, find a number that can not be converted into a cousin by changing a number, and, more difficult still, one smaller than 200.

In the number sequence 2, 10, 12, 16, 17, 18, 19 ... the next number is 200. Why?

**Carlo Frabetti** He is a writer and mathematician, a member of the New York Academy of Sciences. He has published more than 50 scientific dissemination works for adults, children and young people, among them *Damn physics*, *Damn mathematics* or *The big game*. He was a screenwriter *The Cristal ball.*