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A row of five black square tiles is to have a number of its tiles replaced with colored oblong tiles chosen from red (length two), green (length three), or blue (length four). If red tiles are chosen there are exactly seven ways this can be done. If green tiles are chosen there are three ways. And of blue tiles are chosen there are two ways. Assuming that colors cannot be mixed there are 7+3+2=1..
Working from left-to-right if no digit is exceeded by the digit to its left it is called an increasing number; for example, 133468. Similarly if no digit is exceeded by the digit to its right it is called a decreasing number; for example, 66420. We shall call a positive integer that is neither increasing nor decreasing a "bouncy" number; for example, 155349. As \(n\) increases, the proportion of..
In the following equation \(x, \; y,\) and \(n\) are positive integers. \[\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{n}\] For \(n=4\) there are exactly three distinct solutions: \[\begin{split} \dfrac{1}{5} + \dfrac{1}{20} &= \dfrac{1}{4} \\ \dfrac{1}{6} + \dfrac{1}{12} &= \dfrac{1}{4} \\ \dfrac{1}{8} + \dfrac{1}{8} &= \dfrac{1}{4} \end{split}\] What is the least value of \(n\) for which the bumber..