Sept 25 The Dishes Problem

The number of guests must be divisible by 2, 3, 4. You take the number of rice dishes (r) and double it and that gives you the total number of guests. You also get the total number of guests if you triple the number of broth dishes (b) or quadruple the number of meat dishes (m). So 2r=3b=4m. There is also the information that the sum of the dishes is 65, so r+b+m = 65. If we relate the broth and meat to rice, we get b = 2r/3 and m = r/2. Substituting those into the r+b+m expression, we get r + 2r/3 + r/2 = 65 = r(1+ 2/3 + 1/2) = r(6+4+3)/6 = 13r/6 ⇒r = 65(6/13) = 5(6) = 30 ⇒ total number of guests = 2r = 2(30) = 60.

Instead of using latin variables to set up an equation, one can use pictorial diagrams of the food dish that is being described in the question. One may also use words to describe the situation instead of using written equations. However there is an even better method to solve this problem without a system of equations that involves finding the total amount of dishes directly. If we know that 2r = 3b = 4m = n, where n is the total number of dishes, we can say n/2 + n/3 + n/4 = 65 = n(1/2 + 1/3 + 1/4) = n(6+4+3)/12 = 13n/12 ⇒ n = 65(12/13) = 5(12) = 60. This method is much more efficient. 

I believe using examples of puzzles from other cultures will spring interests into the students, so that they are working with something new. They may also feel better to see that mathematicians from their cultures were able to come up with problems as advanced as those studied here. Imagery is helpful for students to better visualize the problem and is more interactive. They're more familiar with whatever interesting character is introduced in the problem, for example a physics question involving R2-D2 catching Luke Skywalker's lightsaber.