Let us take the mass hanging at the end of a spring, bobbing up and down as usual, stop it for a moment and add a rubber band. Thus the upward restoring force is greater than the downward restoring force. When we force the motion, strange things can happen.
The Asymmetric Propeller Revisited
Gillian Saenz, Chris Jackson and Ryan Crumley
Martin Gardner asked in a paper in the Journal (January 1999) if a theorem about a figure produced from triangles would hold if the triangles were replaced with squares. He thought it would, but he couldn't prove it. The authors show that he couldn't prove it. It's false. Too bad!
A Variety of Triangle Inequalities
Herbert Bailey and Yanir Rubinstein
Everybody knows the triangle inequality: if a, b, and c are the sides of a triangle then a + b > c. Here are infinitely many more triangle inequalities. For example, if h is the altitude to side c, then a + b e" (cos A)c + (2 sin A)h where A is any angle from 0 to 90 degrees.
Runs With No Winner in a Lottery
Richard Iltis
Oregon's big bucks lottery started by choosing 6 numbers from 38, changed to six from 42, and is now six from 44. Clearly, the more numbers to choose from, the fewer winners there will be, the bigger the jackpot will be, and the more people will be motivated to gamble. Here is how to find how many fewer winners there will be.
Straightedge Constructions, Given a Parabola
Peter Y. Woo
We do geometrical constructions using a straightedge and a compass. What if we misplaced our compass but had a parabola handy? We could anything we could with s-and-c. And we could draw parabolas.
Meta-Problems in Mathematics
Al Cuoco
(3, 4, 5), (5, 12, 13), (8, 15, 17), . . . are nice right triangles and the author gives a method to find them. Using the same method, we can find nice triangles with a 60-degree angle: (5, 8, 7), (7, 15, 13), (9, 24, 21), . . . . Using the same method, we can find cubic polynomials with integer coefficients, three integral zeros, two integral extrema, and one integral inflection point. There are other applications as well.
Sequences of Chords and of Parabolic Segments Enclosing Proportional Areas
Timothy Feeman and Osvaldo Marrero
The graph of y = x3 and a line tangent to it enclose an area. Draw another tangent line at the point where the first tangent line intersects the curve and repeat. It's known that the areas are proportional. What happens with parabolas? Interesting things!
The Pascal Pyramid
Hans Walser
The big Pascal triangle of binomial coefficients can be tessellated with a lot of little Pascal triangles. The three-dimensional Pascal tetrahedron of multinomial coefficients can be tessellated with a lot of little Pascal tetrahedrons. Here is how, and how to build a physical model.
How many normal lines to a given smooth curve pass through a point not on the curve? It depends, of course. For a parabola, the answer is 1, 2, or 3, and the authors show which points are where. It's a pretty picture.
A Polynomial With a Root Mod m for Every m
Allen Schwenk
Here is a polynomial with small integer coefficients, degree 9, that has no integer zeros but has them (mod m) for every integer m.
On a Theorem of Clay
Hassan Azad and A. Laradji
The theorem of Clay is that the multiplicative group C* is isomorphic to its subgroup S, the unit circle. The authors give a very short proof.
Tangents Without Calculus
Jorge Aaräo
A no-calculus method for finding tangent lines to the graphs of polynomials. It's simple, but not as simple as taking a derivative.