Problem challenge by Ivan Matic
Dear SJMC participants, the title of our next meeting on Wednesday, September 16, is "Counting with Bijections". The leader, Ivan Matic, challenges all of you to try and solve the following four problems before the meeting. Of course, it'll be great if you solve some – or all – of these; but regardless of how many you’ve done there will be lots of fun at the meeting – you will most definitely find interesting new ideas and strategies for tackling similar problems, and many more other problems, as well!
- In how many ways can we distribute 50 apples to 10 students if the first three students have to get at most 5 apples each?
- How many subsets of the set {1, 2, 3, 4, ..., 30} have the property that the sum of the elements of the subset is greater than 232?
- Let n be an odd positive integer. The unit squares of an n by n chessboard are colored alternately black and white, with the four corners colored black. What is the smallest possible value for n for which it is possible to cover all the black squares with non-overlapping figures congruent to an L-shaped figure made of three unit squares?
- Define a domino to be an ordered pair of distinct positive integers. A proper sequence of dominos is a list of distinct dominos in which the first coordinate of each pair after the first equals the second coordinate of the immediately preceding pair, and in which (i, j) and (j, i) do not both appear for any i and j. Let D40 be the set of all dominos whose coordinates are no larger than 40. Find the length of the longest proper sequence of dominos that can be formed using the dominos of D40.