CTK Insights

19 May

Problems with Many Solutions

To an average student (and, perhaps, an average teacher used to teaching from a textbook) it may come as a surprise that there are numerous problems with multiple known solutions. "How come?" - may wonder the average student, "Who would want the drudge of solving a problem whose solution is already known?"

Why, it's a world apart. There are actually people who revel in solving math problems. The reasons are many. One may dislike a known solution or be unaware that one exists. Others consider doing mathematics a plain recreation [Winning Ways, Preface to the 1st edition]. Yet others may come to a solution of a problem by accident. E.g., I believe, Frank Morley was surprised when his study of cardioids inscribed into 3 lines forming a triangle led him to observe three sets of three parallel lines at 60° to each other. Much later he published an elementary solution of the theorem that now bears his name. By now the theorem sports 2 dozen different proofs - several of a very recent vintage

Every proof sheds some light on the nature of a problem, the framework to which it belongs, possible ways to generalize or modify it; every proof makes some of the problem's invisible features visible.

The Pythagorean theorem is probably a champion among various math problems: it counts close to 400 different proofs - some essentially different. The theorem is usually associated with the configuration from Euclid's Elements I.47. Euclid's other proof (VI.31) is less known, but is, by far, more profound bringing to light the fundamental reasons for the validity of the theorem.

I am partial to a relatively recent proof of the Pythagorean theorem by Floor van Lamoen who derived the proof from a very particular case of the isosceles right triangle. The latter was treated by Socrates 2500 years ago. Floor's proof is based on the following diagram:

One of the most beautiful proofs of the Pythagorean theorem was discovered by by Henry Perigal in 1872 and rediscovered by H. E. Dudeney in 1917.

The proof was highly admired for the symmetry of the dissection of two squares involved in the theorem. As far as dissection go, it belongs to a whole family of proofs that stem from two overlaying tessellations of the plane. Just a few weeks ago, Giorgio Ferrarese has observed that such proofs come in pairs, and Perigal's is no exception: he found a proof as symmetrical as Perigal's but that required dissection of a different pair of squares.

One elementary problem with several exquisite solutions and multiple generalizations is on the way of becoming a classic:


Let ABC be an isosceles triangle (AB = AC) with ∠BAC = 20°. Point D is on side AC such that ∠CBD = 50°. Point E is on side AB such that ∠BCE = 60°. Find the measure of ∠CED.

My favorite proof of irrationality of \sqrt{2} depends on the Prime Factorization theorem, but is rather intuitively clear. (This is just one proof out of about 20.) The number of prime factors in the square n^2 of an integer n is double the number of prime factors in n itself. For example, 15=3 \times 5, with two prime factors; 15^2=3\times 3\times5 \times 5, with 4 prime factors. With this understood, assume \sqrt{2}=p/q, where p and q mutually prime integers. Then 2q^2=p^2, where on the left the number of prime factors is odd, whilst on the right it is even. This is impossible.

There is any number of accessible problems with multiple solutions. Perhaps, the infinitude of primes is the most famous one, with Euclid's proof being a mathematical legend, prime example of a beautiful piece of mathematics.

Existence of the orthocenter is another example. Curiously, Euclid never mentioned the orthocenter in the Elements. A general case of intersecting transversals is covered by Ceva's and Menelaus' theorems. Although related, they've been discovered more than 1500 years apart.

In algebra and analysis, we have a fundamental fact of the divergence of the harmonic series, uncountability of the reals, with Cantor himself publishing a couple of proofs (read about the first one).

The great Gauss published three proofs of what we now call the Fundamental Theorem of Algebra and many more followed.

The Arithmetic Mean - Geometric Mean inequality is one of the most basic and oft used theorem in mathematics.

One problem that I have a soft spot for is the problem of 4 travelers:

Four roads on a plane, each a straight line, are in general position so that no two are parallel and no three pass through the same point. Along each road walks a traveler at a constant speed. Their speeds, however, may not be the same. It's known that traveler #1 met with Travelers #2, #3, and #4. #2, in turn, met #3 and #4 and, of course, #1. Please show that #3 and #4 have also met.

That problem has a beautiful solution when embedded in the 3-dimensional space. Other solutions reveal its connection to Ceva's and Menelaus' theorems and, thus, provide a bridge between the two.

In truth, the list of such problems can continue indefinitely, with every new solution adding a little (and sometimes a major) bit to our understanding of the problem itself and of the related pieces of mathematics and its methods. It's a great adventure to seek and find a proof of a problem that makes its validity entirely transparent.

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