Math

A friend claims that she has discovered something interesting. After looking over the lottery 649 draws for the last 5 years, she has noticed that, quite often, when six numbers are drawn, at least two of them are consecutive. For example: 2 21 23 24 25 39 (this has at least two consecutive numbers 23, 24) 5 16 21 34 39 42 (this has no consecutive numbers) 19 21 22 29 34 49 (this has at least two consecutive numbers 21, 22) How often should one expect this to happen in lotto 649?

Computer Science

Consider a sequence of integers of size n. Assume that the only thing that can be done to the sequence is to look at the elements and to reverse the order of the first few consecutive elements.

Using only these operations, is it possible to sort the sequence into non-decreasing order.
If the answer is yes above, how quickly can a sequence containing initially random integers be sorted? We charge 1 each time the first few elements are reversed.

If you like Java, I have some code you can play with. Below is an initial sequence 8 6 9 4 2 If I reverse the order of the first 5 elements I get 2 4 9 6 8 If I now reverse the order of the first 3 elements I get 9 4 2 6 8 If I now reverse the order of the first 2 elements I get 4 9 2 6 8 If I now reverse the order of the first 5 elements I get 8 6 2 9 4 So far, this has cost me 4 reverses, and I have made no progress towards sorting the sequence.

Math

Characterize all squares with vertices on the integer grid. That is, are the only squares with integer endpoints, those that are alligned with the axis? If not, then what do they look like? For example, the following is a square with integer endpoints, it has sides which are alligned with the axis. (0,0),(0,2),(2,2),(2,0)

Computer Science

We say that words x and y are jumbles of eachother if the letters of x can be rearranged to get y. Given a dictionary (file, consisting of one word/line) find the largest group of words, each of which are jumbles of eachother. For example: If the dictionary contained the words arts rats star car arc then arts, rats and star are all jumbles of eachother. The largest group of words, all of which are jumbles of eachother is the above group.