Through hard experience mathematicians have learned to subject even the most 'evident' assertions to rigorous scrutiny, as intuition and facile reasoning can often be misleading. However, errors can slip past the most watchful eye, they are often subtle and difficult to detect; but when found they can teach us a lot and can present a real challenge to straighten out.

This book collects together a mass of such errors, drawn from the work of students, textbooks, and the media, as well as from professional mathematicians themselves. Each of these items is carefully analysed and the source of the error is exposed.

All serious students of mathematics will find this book both enlightening and entertaining.

Mathematical Fallacies, Flaws, and Flimflam In both cases, the equation cannot occur since 3ab(a+b) is not an integer and so the right side is strictly positive. Hence, in all, there are n − 1 solutions to the equation. ♠ To focus ideas, specialize to n = 2. Then on each interval [k1/3 , (k + 1)1/3 ), (1 ≤ k ≤ 7), the function x3 − x3 increases from 0 towards 1, while (x − x )3 increases from 0 to 1 on [1, 2). From the graphs of the two functions, it is straightforward to see that the equation

or new mathematical ideas, reprints and revisions of excellent out-of-print books, popular works, and other monographs of high interest that will appeal to a broad range of readers, including students and teachers of mathematics, mathematical amateurs, and researchers. All the Math That's Fit to Print, by Keith Devlin Circles: A Mathematical View, by Dan Pedoe Complex Numbers and Geometry, by Liang-shin Hahn Cryptology, by Albrecht Beutelspacher Five Hundred Mathematical Challenges, Edward J.

Nazarene University, Kankokee, IL. 12. Tangency by double roots In the days before calculus, one way to check the tangency of two curves with algebraic equations f(x, y) = 0 and g(x, y) = 0 at a common point (a, b) was to eliminate one of the variables from the system of two equations and to check whether the resulting equation in the other variable had a double root corresponding to the common point. As a simple example, y = x2 and y = 2x − 1 represent curves tangent at (1, 1) because x2 = 2x −

28. What the guest contributes to the situation is touched on in Spivak and also discussed by Uri Leron and Mike Eisenberg in their paper, \On a knowledge-related paradox and its resolution," Int. J. Math. Educ. Sci. Technol. 18 (1987) 761{765. In a subtle fashion, Leron and Eisenberg distinguish between facts known to each individual and facts that become public knowledge within the group. I am indebted to a reviewer for directing my attention to the charming children's book, Anno's hat tricks,

variables, the result is: suppose that f(x, y) is defined and differentiable in an open region S, and suppose that x ∂f ∂f +y = pf ∂x ∂y at each point of the region. Then, for (x, y) ∈ S, the relation f(tx, ty) = tp f(x, y) holds in any interval t0 < t < t1 provided t0 ≥ 0, t = 1 is in the interval, and for all t, (tx, ty) ∈ S. To prove this, fix (x, y) and define g(t) = f(tx, ty). Use the hypothesis to prove that tg (t) = pg(t), and then to infer that g(t)t −p is constant (i.e., depends only