![]() ![]() Let s consider the first column of the checkerboard. Let us proceed by contradiction: assume that there exists a coloring such that no subboard has all four corners of the same color. Note that we get a subboard whose corners are the same color if two columns agree on two squares: for example, if column 2 and column 5 both have blue squares in the 1nd and 4th rows, then we d get the following (the squares for which we don t know the colors are colored in light blue): As another example, in the coloring given in the question, columns 1 and 6 both have blue squares in the 3rd and 4th row, which similarly leads to a desired subboard. Here s an example of a coloring: see if you can find a subboard with four corners of the same color! Solution: Let us first discuss how to approach this. Show that there is a subboard all of whose corners are blue or all of whose corners are white. Here s an example of a subboard, with squares shaded in red: Now, suppose that each of the 28 squares is colored either blue or white. A subboard of a checkerboard is a board you can cut-out of the checkerboard by only taking the squares which are between a specified pair of rows and a specified pair of columns. Show that if we take n + 1 numbers from the set then a 1 + a 3 + a 4 + a 5 = a 3 + a 5 + a 7 + a 8 a 1 + a 4 = a 7 + a 8 Thus, this will allow us to find two disjoint subsets of the 10 numbers with the same sum. Then, we discuss what the running time of merge sort would be.1 Pigeonhole Principle Solutions 1. We describe, informally, how we can divide and merge the subproblems, yielding a recurrence relation for the runtime of merge sort. We want to determine an unknown linear order on, and we want to do this by dividing the problem into different subproblems. Ultimately, the lower bound is simplified by using Stirling’s approximation. Then, a theorem is discussed which gives a lower bound on the length of time an arbitrary sorting algorithm can take in the worst-case scenario. This video introduces the sorting problem, and gives an example of a poor sorting algorithm. We ask a set of questions about this list that leads us back to the fair division problem. Suppose, for example, we have a list of n distinct positive integers. In this video we start to develop a framework for understanding the difficulty of a problem. This video introduces big-oh and little-oh notation, and provides a few examples that use these concepts. It is often difficult to say, precisely, what the running times for an algorithm might be. In this video, we consider a list of increasing functions, and ask how quickly they tend to infinity. Within the framework of a mathematics course, we introduce the idea of a problem size and the concept of running time. Here is an easy application of the Pigeon Hole Principle. The Erdős-Szekeres Theorem is introduced, and a proof of this theorem is provided that uses the Pigeon Hole Principle. ![]() In this video, Professor Trotter explains the Erdős number, and tells some stories about this famous mathematician. ![]() This short video introduces the Pigeon Hole Principle, as well as a generalization of it. The Multinomial Theorem gives us an expansion when the base has more than two terms, like in (x 1+x 2+x 3) n. The Binomial Theorem gives us as an expansion of (x+y) n. Here we introduce the Binomial and Multinomial Theorems and see how they are used. How many ways can you rearrange the letters of a string if some of the letters are duplicated? The answer is given by multinomial coefficients. You may want to download the lecture slides that were used for these videos (PDF). ![]()
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