The more ambitious plan may have more chances of success
G. Polya, How To Solve It
Princeton University Press, 1973
Polya continues: ... provided it is not based on a mere pretension but on some vision of the things beyond those immediately present.
Mathematical discovery is seldom a single step process. Often it's indeed the case where answering a more general question is easier than finding an answer to a specific one. Later, on this page, we'll see a collection of examples that I'll be updating from time to time. I would appreciate your mailing me additional examples.
Oftentimes, however, solving a specific problem first may suggest, in a hindsight, a formulation of and a solution to, a more general one. I'll call such a situation
Both circumstances underscore investigative part of doing Mathematics or actually of any kind of problem solving:
- before solving a problem, look around for other formulations, dig your memory for similar known facts
- after a problem has been solved, make this experience an edifying one. Try to stash away as much information as possible. If, in the process, you run into a problem you can't solve right away, go for it - you won't regret it.
I'll call the latter situation
Always look for new problems; especially after successfully solving one - you may get much more than you expected to start with.
Pairs of statements in which one is a clear generalization of another whereas in fact the two are equivalent.
- The Intermediate Value Theorem - The Location Principle (Bolzano Theorem)
- Rolle's Theorem - The Mean Value Theorem
- Existence of a tangent parallel to a chord - existence of a tangent parallel to the x-axis.
- Binomial theorem for (1 + x)n and (x + y)n
- The Maclaurin and Taylor series.
- Two properties of Greatest Common Divisor
- Pythagoras' Theorem and the Cosine Rule
- Pythagoras' Theorem and its particular case of an isosceles right triangle
- Pythagoras' Theorem and Larry Hoehn's generalization
- Combining pieces of 2 and N squares into a single square
- Measurement of inscribed and (more generally) secant angles
- Probability of the union of disjoint events and any pair of events
- Butterfly theorem in orthodiagonal and arbitrary quadrilaterals
- Pascal's theorem in ellipse and in circle
- A triangle is embeddable into a rectangle twice its area and so is any convex polygon
- Brahmagupta's Theorem and Heron's Theorem
There are cases where a more general statement highlights the features of the original problem not otherwise obvious and by doing so spells a solution that works in both cases.
Pairs of statements in which one is a clear generalization of another and the more general statement is not more difficult to prove.
- Bottema's Theorem and McWorter's generalization
- Butterfly theorem
- Butterflies in a Pencil of Conics
- Carnot's theorem and Wallace's theorem.
- Concyclic Circumcenters: A Dynamic View.
- Concyclic Circumcenters: A Sequel.
- Find a plane through a point outside of an octahedron such that the plane bisects the volume of the octahedron - same statement but replace octahedron with a solid with a center of symmetry
- A Fine Feature of the Stern-Brocot tree
- Four Construction Problems
- Lucas' Theorem and its variant
- Matrix Groups
- Napoleon's Theorem and one theorem about similar triangles
- Four Pegs That Form a Square
- On the Difference of Areas
- One Trigonometric Formula and Its Consequences
- Pythagorean Theorem and the Parallelogram Law
- Pythagorean Theorem - General Pythagorean Theorem
- Asymmetric Propeller and Several of Its Generalizations
- A construction problem that combines several apparently unrelated ones
- Two-Parameter Solutions to Three Almost Fermat Equations
- Three circles with centers on their pairwise radical axes
- Square Roots and Triangle Inequality
- Miguel Ochoa's van Schooten is a Slanted Viviani
- Scalar Product Optimization
- Barycenter of cevian trangle generalizes Marian Dinca's criterion.
Pairs of statements in which a more general is directly implied by a more specific one.
- 2D isoperimetric theorem - a similar theorem with a fixed line segment
- Construct a line tangent to two circles - find a tangent from a point to a circle
- Arithmetic mean of N numbers is never less than their geometric mean. N arbitrary and N=2n
- The equation xxx3 = 3 is no more difficult than x3 = 3
- Every convex polygon of area 1 is contained in a rectangle of area 2 because this is true for a triangle
- x1x2 + x2x3 + ... + xn-1xn ≤ 1/4, for x1 + x2 + ... + xn = 1, privided it is true for n = 2.
- Vietnamese Extension of a Japanese Theorem. Property of Two Pencils of Parallel Lines
Problems that allow a meaningful generalization.
- In a plane, given 3n points. Is it possible to draw n triangles with vertices at these points so that no two of them have points in common?
- Lines in a triangle intersecting at a common point. Ceva's theorem.
- 5109094x171709440000 = 21!, find x.
- Given a 1x1 square. Is it possible to put into it not intersecting circles so that the sum of their radii will be 1996?
- Criteria of divisibility by 9 and 11.
- The game of Fifteen and Puzzles on Graphs
- Weierstrass Product Inequality
- Fermat's Little Theorem and Euler's Theorem
Problems that allow more than one generalization.
- Pythagorean Theorem
- Napoleon's Theorem
- Fermat Point and Generalizations
- A problem in extension fields
- Butterfly theorem
- A System of Equations Begging for Generalization
Thus we see that generalization is quite useful and often enjoyable. It's a great vehicle for discovering new facts. However, if unchecked, generalization may lead to erroneous results. I'd call such situations
Following are a few examples where attempts to generalize lead one astray. (There eventually will be a few.)
I have allowed myself to swerve from Pólya's maxim: The more ambitious plan may have more chances of success. "The more ambitious" may not necessarily mean "the more general". Following are a few examples where a "more powerful, stronger" statement comes out easier to prove:
1/2·3/4·5/6· ... ·99/100 < 1/10.
- A Low Bound for Inequality 1/2·3/4·5/6· ... ·(2n - 1)/2n.
(1 + 1-3)(1 + 2-3)(1 + 2-3)...(1 + n-3) < 3.
1 + 2-2 + 3-2 + 4-2 + ... + n-2 < 2.
1/(n+1) + ... + 1/2n < 25/36.
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