And Other Curious Questions

Pride in craftsmanship obligates the mathematicians of one generation to dispose of the unfinished business of their predecessors.

Two familiar irrational numbers are it (3.141 . . .), the ratio of the circumference of a circle to its diameter, and e (2.718 . . .), the base of natural logarithms. Each has a nonrepeating decimal fraction. Both tt and e are also transcendental numbers, that is, numbers that are not algebraic. Specifically, a transcendental number is an irrational number that is not the root of an algebraic equation with rational coefficients. Is the sum of tt and e transcendental? No mathematician knows if the sum is even irrational.

One might suppose that any two numbers with infinite, nonrepeating decimal fractions would necessarily have a sum with a nonrepeating (therefore irrational) decimal fraction. This is not the case. The difference between it and 7, for instance, is another transcendental. It is easy to compute. Represent 7 as 6.999 . . . , then subtract tt (3.14159 . . .) to obtain the transcendental number 3.858407. . . . The sum of these two transcendental obviously is 6.999 . . . , or 7.

It seems unlikely, but until someone proves otherwise it and e could be related by a curious unknown formula that would give their sum a repeating (rational) decimal fraction with a very long period, say one of more than a billion digits. It also is not known if ire, tt11, e1 or ire are irrational. It has been shown, however, that e* is transcendental, and it is easy to prove that at least one of the two numbers, -ne and (ir + e), is transcendental. The unanswered questions about tt and e are among hundreds of problems that are ridiculously simple to state but so difficult and deep that long-lasting fame awaits the first person to solve them.

It is not easy to distinguish significant unsolved problems from trivial ones. In A Mathematician's Apology, G. H. Hardy characterized a significant problem as being one connected to such a large complex of other mathematical ideas that when it is solved, it leads to important advances in mathematics and perhaps in science as well. An example of an essentially trivial but extremely difficult question is: If two people play the best possible checker game, will it end in a draw, a victory for the player who makes the first move or a victory for the player who makes the second move? A computer, given enough time, will probably work out the answer one day. When it does, the solution is unlikely to lead to any breakthroughs in mathematics or science. On the other hand, settling Fermat's last theorem would open all kinds of barred doors. (Please do not send me proofs. I am incapable of spotting flaws and always return them unread.)

There are dozens of unsolved map-coloring problems that, although they may not be as profound as the recently solved four-color theorem, are by no means trivial. Here is a notorious one given in C. Stanley Ogilvy's new revision of Tomorrow's Math, a splendid collection of unsolved problems for amateurs. What is the minimum number of colors needed to color the plane in such a way that any pair of points a unit distance apart are in regions of different colors? The question was first raised 20 years ago by Paul Erdos, a prolific inventor of problems.

That such a map must have at least four colors was cleverly established by Leo Moser with the diagram in Figure 118. Each edge of this graph has a length of one unit. Imagine that the graph is placed anywhere on a plane in which the problem is solved with only three colors. If vertex a is on red, say, then b and c must be on the other two colors, and g also must be red. Similarly, d and e must be on the other two colors and / must be red. Now, however, we have contradicted our assumption, because / and g, which are a unit distance apart, are both on red. At least four colors are therefore necessary.

Ogilvy's book has a proof that seven different colors are enough [see Figure 119]. The numbers give a repeating color pattern for a hexagonal tesselation of the plane, each hexagon b b

Leo Moser's graph for proving four-color necessity

Figure 119

Figure 119

slightly less than one unit from corner to opposite corner. The gap remaining to be closed is a big one. Do such maps of four, five or six colors exist? No one yet knows.

There is an unusual class of unsolved arithmetic problems that, to use a computer-science term, we can call "looping" problems. A series of integers is generated according to a rule. One then asks if the series always enters one or more loops in which a finite set of integers keeps repeating cyclically. For example, start with any positive integer. Halve it if it is even; triple it and add 1 if it is odd. Keep repeating this procedure until the series loops in the cycle 2, 1, 4, 2, 1, 4, ... . (Sample: 3, 10, 5, 16, 8, 4, 2, 1, . . . .) But does the series always enter the 2, 1, 4 loop? No one has proved that it does, nor has a counterexample been found.

Since Ogilvy revised his book a group of workers in the Artificial Intelligence Laboratory at the Massachusetts Institute of Technology have computer-tested all positive integers to 60,000,000 without finding an exception. They also discovered that if the rule 3n+ 1 (for odd integers) is replaced by 3n — 1, the result, in absolute values, is the same as starting with a negative integer and following the old rules. In this case all negative integers to — 100,000,000 were found to enter one of three loops with the following absolute values:

0 0

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