OK, so we’ve solved the four-color map theorem and Fermat’s Last Theorem–but what about these gems that I came across:

The integral cube, in which all sides and all diagonals are integers.
The palindromic numbers conundrum, in which a number is added to its reverse digits over a period of iterations, it will result in a palindromic number. For example:

I’m not sure if I understand what you mean about the integral cube. I’ve never come across the phrase except when it means a cube that is also an integer, as 9, 27, etc.

The diagonal of a cube is the length of a side times the square root of three. If the length of a side is an integer and the square root of three is irrational (see proof) then the diagonal of the cube must also be irrational. Likewise the diagonals of the faces are irrational because the diagonal is the square root of two times the length of the side. (See this proof that the square root of two is irrational.)

I believe the problem is properly called the integral
brick problem. Construct a rectangular solid, or
parallelepiped if you will, so that the length, width,
height, all 3 face diagonals and the interior diagonal
are integers. Never solved and never proved impossible.

Leonhard Euler found the smallest brick with integral
edges and face diagonals has edges 44, 117 and 240.
If all values are integral except a face diagonal,
the smallest brick has edges of 104,153, and 672.
John Leech, a British mathematician, ‘suspects’ the
smallest brick where only an edge is non-integral is
one with edges of 7800, 18720 and sqrt(211773121).

I believe the palindrome problem is unsolved as well.

Actually, the most important unsolved problem in math right now is the Riemann hypothesis. The Riemann zeta function is:

1^(-x) + 2^(-x) + 3^(-x) + 4^(-x) + …

The Riemann hypothesis says that this is equal to zero only when the imaginary part of x is 1/2. A lot more will be known about the primes when this question is settled.

Then there’s the continuum hypothesis, which says

2^(aleph-0) = aleph-1.

This was listed by Hilbert at the turn of the century as the most important problem to be solved in mathematics. It turned out to be undecidable under the standard axioms of set theory.

Actually, the reason I bring that last one up is that this year I heard I new axiom has been introduced (Woodin’s), which apparently has some intuitive appeal and implies that the continuum hypothesis is false. There’s speculation as to whether or not this axiom will gain acceptance in the mathematical community. Does anyone know what this axiom says?

What about Goldbach’s conjecture (every even number is the sum of two primes), and the Collatz or 3n+1 problem (start with an integer and either divide it by two if it is even, or multiply by three and add one–repeat until the sequence repeats. Do you eventually arrive at 1.)?

Here’s one from my calculus book…
Limacons (a type of polar equation) and circles have what is known as an equichordal point where all chords passing through the point have equal lengths (the center of the circle, obviously; somewhat more complicated for a limacon).

The unsolvable problem: Is there a plane region that has 2 equichordal points?

The book adds the caveat that a correct proof for this would make you instantly famous, but you should be sure to work the problems at the end of the chapter before trying it.

How about the Twin Primes conjecture (i.e., that there are an infinite number of pairs of primes of difference 2, such as 5,7 or 197, 199), or the conjectures that there are no odd perfect numbers (the sum of the factors equalls the nuber, such as 6 or 28), or an infinite amount Merseinne primes (primes of the form 2^n - 1)? Personally, I tend to think that the greater Goldbach conjecture is the biggest, since it’s familiar to so many folks and so counterintuitive.

As for the palindrome problem, how long does the process go on starting with 98 (or 89)? I actually had nightmares about that one back in fourth grade when we learned about those… adding numbers in my sleep and muttering “not a palindrome”…

“all numbers less than 10,000 have been tested. Every one becomes a palindrome in a relatively small number of steps (of the 900 3-digit numbers, 90 are palindromes to start with and 735 of the remainder take less than 5 reversals and additions to yield a palindrome).Except, that is, for 196.”

196 has been reversed and added until it reaches 2 million digits and has not become a palindrome. The program that was used to compute this as well as the 2 million digits are available from

Oops, I should have said real part, not imaginary part.

Also with the Riemann zeta function, it’s exact values are known when x=2,4,6,… (the values can be expressed in terms of pi), but no exact values have been found when x=3,5,7,…

Um, I’m not a mathematician, but then again, I’m not a physicist and sometimes I think about time paradoxes. To the question Is there a plane figure with two equichordial points? I’d guess no, because any two points define a line and the line passing through any two equichordial points would have to be a different length from any of the other lines passing through either of said points. …wouldn’t it?

Well, I wrote a program for fun to do the number cruching, and 196 doesn’t converge to a palindrome in 7502000 iterations. At that many iterations, I have 3105883 digits (this took about 6 days). Unfortunately, I can’t find a contact for that guy on the web page to give him the numbers. Sigh.

I did some experiments random numbers with bases other than 10, and they, too, ended up in a palindrome.

I wonder if they have their own asymptomatic (that is, non-palindromic) numbers like base 10’s 196. (At least, if it does turn into a palindrome, it doesn’t anytime soon.) Or, maybe the number 196 never (within reason) turns into a palindrome regardless of what base is being used. (By 196, I mean whatever numerals are used to represent that number in whatever base.)

It says the problems include “the following equations, named for the mathematicians who postulated them: the Riemann Hypothesis, the Poincare Conjecture, the Hodge Conjecture, the Birch and Swinnerton-Dyer Conjecture, Navier-Stokes Equations, the Yang-Mills Theory and the P versus NP Problem.”

What I want to know is, who was P and NP, and why were they at odds?