Math Horizons September 2014 : Page 28

3 V R 5 R 7 References 1. E. Berlekamp and J. Buhler, “Puzzles” Column, Emissary (newsletter of the Mathematical Sciences Research Institute), Spring 2013. 2. A. riedland, Math and Logic, New York: Dover, 1971. 3. G. Galperin, Playing pool with (The number from a billiard point of view), Regular and Chaotic Dynamics, 8 (2003): 375–394. 4. J. P. Hutchinson and S. Wagon, A forbidden subgraph characterization of in nite graphs having nite genus, in Graphs and Applications, Proc. of the First Colo. Symposium on Graph Theory , . Harary, J. Maybee, eds., New York: Wiley, 1985. 5. V. Klee, The generation of affne hulls, Acta Sci. Math. (Szeged) 24 (1963): 60 – 81. 6. J. Konhauser, D. Velleman, and S. Wagon, Which Way Did the Bicycle Go? , Washington, DC: MAA, 1996. 7. . Rubin, Problem 729, J. Rec. Math. 11 (1979): 128; 12 (1980): 145. 8. S. Wagon, The Cake Icing Puzzle, from the Wolfram Demonstrations Project, 9. S. Wagon, Macalester College Problem of the Week, . 10. P. Winkler, Mathematical Mind-Benders, Wellesley, MA: A K Peters, 2007. Stan Wagon recently retired from Macalester College. His books include The Banach-Tarski Paradox , Math-ematica in Action , and VisualDSolve . His interests include geometric snow sculpture, mountaineering, and mushroom hunting. He is a founding editor of UltraRunning magazine, but now finds that covering long distances is much easier on skis than in running shoes. Email: 2 R 3 1 R 1 R 4 R 2 R 12 R 8 0 1 R 6 R 10 2 0 2 cover the PV-plane. 4 P Figure 4. The rectangles strip’s left boundary (for example, the red rectangle in gure 3). to cover the ver-Use the rectangles tical strip corresponding to and (see gure 4). We know the rectangles will cover the strip because the sum of the heights diverges. Likewise, use to cover the vertical the rectangles and Then strip corresponding to use the rectangles and to cover the strip Continue in this way, using collec-to cover the tions of rectangles plane via vertical strips of width two. (There are other ways to cover the plane, but this is nicely simple.) to decide where to At time n , use rectangle shoot. In particular, if is the lower left corner then re at the real number This of shot will strike ships whose initial conditions are in in particular, it will the strip determined by strike ships whose initial conditions are in Because the rectangles cover the PV-plane, any ship will be hit. [9, problem 1164] 10. Shocking Collisions. There are 314,159 col-the collision lisions. Moreover, if B ’s mass is count is the integer corresponding to the rst The proof requires a reasonable, but un-digits of proved, assumption about the digits of . n 28 September 2014 : : Math Horizons : :

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