Dragons and Kasha

Abstract

Kasha-eating dragons introduce advanced mathematics. The goal of this paper is twofold: to entertain people who know advanced mathematics and inspire people who don’t. Suppose a four-armed dragon is sitting on every face of a cube. Each dragon has a bowl of kasha in front of him. Dragons are very greedy, so instead of eating their own kasha, they try to steal kasha from their neighbors. Every minute every dragon extends four arms to the four neighboring faces on the cube and tries to get the kasha from the bowls there. As four arms are fighting for every bowl of kasha, each arm manages to steal one-fourth of what is in the bowl. Thus each dragon steals one-fourth of the kasha of each of his neighbors, while at the same time all of his own kasha is stolen. Given the initial amounts of kasha in every bowl, what is the asymptotic behavior of the amounts of kasha? Why do these dragons eat kasha? Kasha (buckwheat porridge) is very healthy. But for mathematicians, kasha represents a continuous entity. You can view the amount of kasha in a bowl as a real number. Another common food that works for this purpose is soup, but liquid soup is difficult to steal with your bare hands. We do not want to s soup spilled all over our cube, do we? If kasha seems too exotic, you can imagine less exotic and less healthy mashed potatoes. How does this relate to advanced mathematics? For starters, it relates to linear algebra [3]. We can consider the amounts of kasha as six real numbers, as there are six bowls, one on each of the six faces of the cube. We can view this six-tuple that represents kasha at each moment as a vector in a six-dimensional vector space of possible amounts of kasha. To be able to view the amounts of kasha as a vector, we need to make a leap of faith and assume that negative amounts of kasha are possible. I just hope that if my readers have enough imagination to envision six four-armed dragons on the faces of the cube, then they can also imagine negative kasha. The bowl with 2 pounds of kasha means that if you put two pounds of kasha into this bowl, it becomes empty. For those who wonder why dragons would fight for negative kasha, this is how mathematics works. We make unrealistic assumptions, solve the problem, and then hope that the solution translates to reality anyway.