Table Of Content

INDEX TO VOLUME 64 AUTHORS Kleiner, Israel, Rigor and Proof in Mathematics: A Historical Perspective, 291-314. Althoen, S.C., and C.B. Lacampagne, Tetrahedral Knuth, Donald, see Gardner, Martin. Numbers as Sums of Square Numbers, 104-108. Koster, D.W., see Killgrove, R.B. Ayoub, Ayoub B., Proof without Words: The Kung, Sidney, Proof without Words: sin(x+y)=sin x Reflection Property of a Parabola, 175. cos y forx+y<z, 97. Baker, John A., Plane Curves, Polar Coordinates, Lacampagne, C.B., see Althoen, S.C. and Winding Numbers, 75-91. Lehel, Jené, Proof without Words: The Sum of Odd Bilodeau, G.G., Generating Polynomials All of Numbers, 103. Whose Roots are Real, 263-270. Logothetti, Dave, Proof without Words, 114. Botsko, Dennis C., A Fundamental Theorem of , see Chakerian, Don. Calculus that Applies to All Riemann Integrable Massell, Paul B., Volumes of Cones, Paraboloids, Functions, 347-348. and Other “Vertex Solids", 122-123. Campbell, Douglas M., and David Radford, Tree McKenney, James E., Finding the Volume of an Isomorphism Algorithms: Speed vs. Clarity, 252- Ellipsoid Using Cross-Sectional Slices, 32-34. 261. Moen, Courtney, Infinite Series with Binomial Chakerian, Don, and Dave Logothetti, Cube Slices, Coefficients, 53-55. Pictorial Triangles, and Probability, 219-241. Murphy, Gratian, see Danzer, Ludwig. Clevenson, M. Lawrence, and Wiiliam Watkins, Nelsen, Roger, B., Proof without Words: An Majorization and the Birthday Inequality, 183-188. Arctangent Identity and Series, 241. Cooke, William P., From Fences to Hyperboxes and » Proof without Words: Integration by Parts, Back Again, 340-342. 130. Danzer, Ludwig, Grattan Murphy, and John Reay, , Proof without Words: Sums of Reciprocals Translational Prototiles on a Lattice, 3-12. of Triangular Numbers, 167. Dodd, Fred, and Rhodes Peele, Some Counting Nelsen, Roger B., and Harvey Schmidt, Jr., Chains Problems Ii ig the Multi ial Exp in Power Sets, 23-31. 115-122. Nissen, Phillip, and John Taylor, Running Clubs—A Drobot, Vladimir, and Lawrence J. Wallen, Combinatorial Investigation, 39-44. Asymptotic Iteration, 176-180. Peele, Rhodes, see Dodd, Fred. Du, D.Z., and F.K. Hwang, Steiner Minimal Trees Pooley, Sam, Proof without Words: Algebraic Areas, on Chinese Checkerboards, 332-339. 138. Fife, James H., The Kuratowski Closure-Complement Rabinowitz, Stanley, see Gilbert, Peter. Problem, 180-182. Radford, David, see Campbell, Douglas M. Gallant, Charies D., Proof without Words: Ramankutty, P., The Complementary Function and Comparing B* and A® for A<B, 31. the General Solution, 124-130. Gallian, Joseph A., Assigning Driver's License Ratering, Steven, An Interesting Subset of the Highly Numbers, 13-22. Composite Numbers, 343-346. Galovich, Steven, and Jeremy Resnick, Representing Reay, John, see Danzer, Ludwig. Primes by Binary Quadratic Forms, 34-38. Resnick, Jeremy, see Galovich, Steven. Gardner, Martin, Richard K. Guy, Donald Knuth, Rigby, J.F., Cycles and Tangent Rays, 155-167. and Lee Sallows, Serial Isogons of 90 Degrees, , Napoleon, Escher, and Tessellations, 242- 315-324. 246. Gau, Yihnan David, Proof without Words, 339, Russell, Dennis C., Another Eulerian-Type Proof, Getu, Seyoum, Eval. ig Deter via 349. Generating Functions, 45-53, Sallows, Lee, see Gardner, Martin. Gilbert, Peter, and Stanley Rabinowitz, A Nonlinear Schaumberger, Norman, A Coordinate Approach to the AM-GM Inequality, 273. Recurrence Yielding Binary Digits, 168-171. Guy, Richard K., see Gardner, Martin. Schmidt, Je., Harvey, see Nelsen, Roger B. Hwang, F.K., see Du, D.Z. Schulz, William C., Cubics with a Rational Root, Jepsen, Charles H., Packing a Box with Bricks, 92- 172-175. 97 Schwenk, Allen J., Which Rectangular Chessboards Khamis, H.J., Buffon’s Needle Problem on Radial Have a Knight's Tour?, 325-332. Lines, 56-58. Solomon, Frederick, Monte Carlo Simulation of Killgrove, R.B., and D.W. Kosier, Regular Polygons Infinite Series, 188-196. with Rational Area or Perimeter, 109-114. Stone, A.P., Power Series Expansions for Klamkin, Murray S., On Some Symmetric Sets of Trigcivometric Functions via Solutions to Initial Unit Vectors, 271-273. Value Problems. 247-251. 364 INDEX TO VOLUME 64 AUTHORS Kleiner, Israel, Rigor and Proof in Mathematics: A Historical Perspective, 291-314. Althoen, S.C., and C.B. Lacampagne, Tetrahedral Knuth, Donald, see Gardner, Martin. Numbers as Sums of Square Numbers, 104-108. Koster, D.W., see Killgrove, R.B. Ayoub, Ayoub B., Proof without Words: The Kung, Sidney, Proof without Words: sin(x+y)=sin x Reflection Property of a Parabola, 175. cos y forx+y<z, 97. Baker, John A., Plane Curves, Polar Coordinates, Lacampagne, C.B., see Althoen, S.C. and Winding Numbers, 75-91. Lehel, Jené, Proof without Words: The Sum of Odd Bilodeau, G.G., Generating Polynomials All of Numbers, 103. Whose Roots are Real, 263-270. Logothetti, Dave, Proof without Words, 114. Botsko, Dennis C., A Fundamental Theorem of , see Chakerian, Don. Calculus that Applies to All Riemann Integrable Massell, Paul B., Volumes of Cones, Paraboloids, Functions, 347-348. and Other “Vertex Solids", 122-123. Campbell, Douglas M., and David Radford, Tree McKenney, James E., Finding the Volume of an Isomorphism Algorithms: Speed vs. Clarity, 252- Ellipsoid Using Cross-Sectional Slices, 32-34. 261. Moen, Courtney, Infinite Series with Binomial Chakerian, Don, and Dave Logothetti, Cube Slices, Coefficients, 53-55. Pictorial Triangles, and Probability, 219-241. Murphy, Gratian, see Danzer, Ludwig. Clevenson, M. Lawrence, and Wiiliam Watkins, Nelsen, Roger, B., Proof without Words: An Majorization and the Birthday Inequality, 183-188. Arctangent Identity and Series, 241. Cooke, William P., From Fences to Hyperboxes and » Proof without Words: Integration by Parts, Back Again, 340-342. 130. Danzer, Ludwig, Grattan Murphy, and John Reay, , Proof without Words: Sums of Reciprocals Translational Prototiles on a Lattice, 3-12. of Triangular Numbers, 167. Dodd, Fred, and Rhodes Peele, Some Counting Nelsen, Roger B., and Harvey Schmidt, Jr., Chains Problems Ii ig the Multi ial Exp in Power Sets, 23-31. 115-122. Nissen, Phillip, and John Taylor, Running Clubs—A Drobot, Vladimir, and Lawrence J. Wallen, Combinatorial Investigation, 39-44. Asymptotic Iteration, 176-180. Peele, Rhodes, see Dodd, Fred. Du, D.Z., and F.K. Hwang, Steiner Minimal Trees Pooley, Sam, Proof without Words: Algebraic Areas, on Chinese Checkerboards, 332-339. 138. Fife, James H., The Kuratowski Closure-Complement Rabinowitz, Stanley, see Gilbert, Peter. Problem, 180-182. Radford, David, see Campbell, Douglas M. Gallant, Charies D., Proof without Words: Ramankutty, P., The Complementary Function and Comparing B* and A® for A<B, 31. the General Solution, 124-130. Gallian, Joseph A., Assigning Driver's License Ratering, Steven, An Interesting Subset of the Highly Numbers, 13-22. Composite Numbers, 343-346. Galovich, Steven, and Jeremy Resnick, Representing Reay, John, see Danzer, Ludwig. Primes by Binary Quadratic Forms, 34-38. Resnick, Jeremy, see Galovich, Steven. Gardner, Martin, Richard K. Guy, Donald Knuth, Rigby, J.F., Cycles and Tangent Rays, 155-167. and Lee Sallows, Serial Isogons of 90 Degrees, , Napoleon, Escher, and Tessellations, 242- 315-324. 246. Gau, Yihnan David, Proof without Words, 339, Russell, Dennis C., Another Eulerian-Type Proof, Getu, Seyoum, Eval. ig Deter via 349. Generating Functions, 45-53, Sallows, Lee, see Gardner, Martin. Gilbert, Peter, and Stanley Rabinowitz, A Nonlinear Schaumberger, Norman, A Coordinate Approach to the AM-GM Inequality, 273. Recurrence Yielding Binary Digits, 168-171. Guy, Richard K., see Gardner, Martin. Schmidt, Je., Harvey, see Nelsen, Roger B. Hwang, F.K., see Du, D.Z. Schulz, William C., Cubics with a Rational Root, Jepsen, Charles H., Packing a Box with Bricks, 92- 172-175. 97 Schwenk, Allen J., Which Rectangular Chessboards Khamis, H.J., Buffon’s Needle Problem on Radial Have a Knight's Tour?, 325-332. Lines, 56-58. Solomon, Frederick, Monte Carlo Simulation of Killgrove, R.B., and D.W. Kosier, Regular Polygons Infinite Series, 188-196. with Rational Area or Perimeter, 109-114. Stone, A.P., Power Series Expansions for Klamkin, Murray S., On Some Symmetric Sets of Trigcivometric Functions via Solutions to Initial Unit Vectors, 271-273. Value Problems. 247-251. 364 VOL. 64, NO. 5, DECEMBER 1991 365 Taylor, John, see Nissen, Phillip. On Some Symmetric Sets of Unit Vectors, Murray S. Tingley, Daryl, Complements of Linear Subspaces, Klamkin, 271-273. 98-103. Packing a Box with Bricks, Charles H. Jepsen, 92- Troesch, B.A., Archimedes’ Method for the 97. Reflections on the Ellipse, 262-263. Plane Curves, Polar Coordinates, and Winding Wallen, Lawrence J., see Drobot, Vladimir. Numbers, John A. Baker, 75-91. Watkins, William, see Clevenson, M. Lawrence. Power Series Expansions for Trigonometric White, Arthur, Limerick, 91. Functions via Solutions to Initial Value Problems, A.P. Stone, 247-251. TITLES Proof without Words, Yihnan David Gau, 339. Proof without Words, Dave Logothemti, 114. Another Eulerian-Type Proof, Dennis C. Russell, Proof without Words: Algebraic Areas, Sam Pooley, 349. 138. Archimedes’ Method for the Reflections on the Proof without Words: Arctangent Identity and Series, Ellipse, B.A. Troesch, 262-263. An, Roger B. Nelsen, 241. Assigning Driver’s License Numbers, Joseph A. Proof without Words: Comparing BA and A® for Gallian, 13-22. A<B, Charles D. Gallant, 31. Asymptotic Iteration, Viadimir Drobot and Lawrence Proof without Words: Integration by Parts, Roger B. J. Wallen, 176-180. Nelsen, 130. Buffon’s Needle Problem on Radial Lines, H.J. Proof without Words: Reflection Property of the Khamis, 56-58. Parabola, The, Ayoub B. Ayoub, 175. Chains in Power Sets, Roger B. Nelsen and Harvey Proof without Words: sin(x+y)=sin x cos y + cosx Schmidt, Jr., 23-31. sin y for x+y<x, Sidney H. Kung, 97. Complementary Function and the General Solution, Proof without Words: Sum of Odd Numbers, The, The, P. Ramankutty, 124-130. Jen6 Lehel, 103. Complements of Linear Subspaces, Daryl Tingley, Proof without Words: Sum of Reciprocals of 98-103. Triangular Numbers, Roger B. Nelsen, 167. Coordinate Approach to the AM-GM Inequality, A, Regular Polygons with Rational Area or Perimeter, Norman Schaumberger, 273. R.B. Killgrove and D.W. Koster, 109-114. Cube Slices, Pictorial Triangles, and Probability, Representing Primes by Binary Quadratic Forms, Don Chakerian and Dave Logothetti, 219-241. Steven Galovich and Jeremy Resnick, 34-38. Cubics with a Rational Root, William C. Schulz, 172- Rigor and Proof in Mathematics: A Historical 175. Perspective, Israel Kleiner, 291-314. Cycles and Tangent Rays, J.F. Rigby, 155-167. Running Clubs—A Combinatorial Investigation, Evaluating Determinants via Generating Functions, Phillip Nissen and John Taylor, 39-44. Seyoum Getu, 45-53. Serial Isogons of 90 Degrees, Martin Gardner, Finding the Volume of an Ellipsoid Using Cross- Richard K. Guy, Donald Knuth, and Lee Sallows, Sectionai Slices, James E. McKerney, 32-34. 315-324. From Fences to Hyperboxes and Back Again, Some Counting Problems Involving the Multinomial William P. Cooke, 340-342. Expansion, Fred Dodd and Rhodes Peele, 115-122. Fundamental Theorem of Calculus that Applies to All Steiner Minimal Trees on Chinese Checkerboards, Riemann Integrable Functions, A, Michael W. D.Z. Du and F.K. Hwang, 332-339. Botsko, 347-348. Tetrahedral Numbers as Sums of Square Numbers, Generating Polynomials All of Whose Roots Are S.C. Althoen and C.B. Lacampagne, 104-108. Real, G.G. Bilodeau, 263-270. Translational Prototiles on a Lattice, Ludwig Danzer, Infinite Series with Binomial Coefficients, Courmey Granan Murphy, and John Reay, 3-12. Moen, 53-55. Tree Isomorphism Algorithms: Speed vs. Clarity, Interesting Subset of the Highly Composite Numbers, Douglas M. Campbell and David Radford, 252- An, Steven Ratering, 343-346. 261. Kuratowski Closure-Complement Problem, James H. Volumes of Cones, Paraboloids, and Other "Vertex Fife, 180-182. Solids”, Paul B. Massell, 122-123. Limerick, Arthur White, 91. Which Rectangular Chessboards Have a Knight's Majorization and the Birthday Inequality, M. Tour?, Allen J. Schwenk, 325-332. Lawrence Clevenson and William Watkins, 183-188. Monte Carlo Simulation of Infinite Series, Frederick PROBLEMS Solomon, 188-196. Napoleon, Escher, and Tessellations, J.F. Rigby, P, S, Q, and C refers to Proposals, Solutions, 242-246. Quickies, and Comments, respectively. Page Nonlinear Recurrence Yielding Binary Digits, A, numbers are given in parentheses. Thus, P1338(55) Peter Gilbert and Stanley Rabinowitz, 168-171. 366 MATHEMATICS MAGAZINE refers to proposal number 1338, which appears on Konetny, Vaclav, $1354(277). page 55. Kotiah, Thoddi C.T., P1380(274). Kung, Sidney H., P1365(60). February: P1363-1367; Q772-774; S1338-1342. Lee, Eugene, $1351(203). April: P1368-1372; Q775-777; $1343-1347. Lindquist, Norman F., §13501(199). June: P1373-1377; Q778-780; $1348-1352. Lord, Nick, P1372(132). October: P1378-1382; Q781-783; $1353-1357. Luthar, R.S., P1370(132). December: P1383-1387; Q784-785; $1358-1362. Martin, Reiner, $1347(136). , $13611354). Athanasiadis, Christos, $1355(278). Moews, David, and Michael Reid, P1374(197). , $1356(278). Morris, Howard, P1366(60). Beasley, Brian D., $1339(63). , P138550). Bencze, Mihaly, P1369(131). Noland, Hugh, Q782(275). , P1381(275). Penrice, Stephen G., P1378(274). Bivens, Iri C., Q772(60). Poole, David G., $13501(200). Brunson, Barry, $1342(65). Rabinowitz, Stanley, P1364(60). Caldwell, Chris, Q776(132). Q781(275). Calla, David, $1344(134). Rebman, Ken, Q778(198). » $1357(279). Reid, Michael, see Moews, David. , $136101G54). Sadoveanu, Ioan, $1359(352). Canning, Eric, and Marion B. Smith, P1376(197). Schaumberger, Norman, Q780(198). Christos, Athanasiadis, $1349(199). Schmidt, Frank W., Q773(61). Schramm, Mike, see Farrell, Kevin. Con Amore Problem Group, $1343(132). » $135201(205). Schwenk, Allen J., P1368(131). 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Bartlett, Albert Allen, 286. , P1379(274). Kedlaya, Kiran, 286. : Klamkin, Murray S., Q774(61). Miller, Andreas, 361. » C761(65). Robertson, John P., 361. , Q779(198). Sondow, Jonathan D., 286. , C13$4(277). Whittsitt, Stephen J., 144. , C774(356). 19th Annual U.S.A. Mathematical Olympiad Klamkin, Murray S., and Andy Liu, Q777(132). Problems and Solutions, 211. » Q783(275). 31st Annual International Mathematical Olympiad Problems and Winners, 213.