# Simple unsolved math problem, 8

Sylver coinage is a game for 2 players invented by John H. Conway.

The two players take turns naming positive integers that are not the sum of non-negative multiples of any previously named integers. The player who is forced to name 1 loses.

James Joseph Sylvester proved the following fact.

Lemma: If a and b are relatively prime positive integers, then (a – 1)(b – 1) – 1 is the largest number that is not a sum of nonnegative multiples of a and b.

Therefore, if a and b have no common prime factors and are the first two moves, this formula gives an upper bound on the next number that can still be played.

R. L. Hutchings proved the following fact.

Theorem: If the first player selects any prime number $p>3$ as a first move then he/she has a winning strategy.

Very little is known about the subsequent winning moves. That is, a winning strategy exists but it’s not know what it is!

Unsolved problem:Are there any non-prime winning opening moves in Sylver coinage?

For further info, Sicherman maintains a Sylver coinage game webpage.

# Simple unsolved math problem, 7

Everyone’s heard of the number $\pi =$ 3.141592…, right?

Robert Couse-Baker / CC BY http://2.0 / Flickr: 29233640@N07

And you probably know that $\pi$ is not a rational number (i.e., a quotient of two integers, like 7/3). Unlike a rational number, whose decimal expansion is eventually periodic, if you look at the digits of $\pi$ they seem random,

3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482…

But are they really? No one really knows. There’s a paper that explores the statistics of these digits using the first 22.4 trillion digits of $\pi$. Does any finite sequence of k digits (say, for example, the 4-digit sequence 2016) occur just as often as any other sequence of the same length (say, 1492), for each k? When the answer is yes, the number is called ‘normal.’ That is, a normal number is a real number whose infinite sequence of digits is distributed uniformly in the sense that each digit has the same natural density 1/10, also all possible k-tuples of digits are equally likely with density 1/k, for any integer $k>1$.

The following simple problem is unsolved:

Conjecture: $\pi$ is normal.

# Simple unsolved math problem, 6

If you know a little point-set topology, below is an unsolved math problem whose statement is relatively simple.

Probably everyone has at least seen the Mandelbrot set in some form, as it’s a popular object of mathematical artists. Here’s a picture from Wikipedia:

The formal definition is as follows. Let $f_c (z)=z^2+c$, where $c\in \mathbb{C}$ is a complex number. The Mandelbrot set $X$ is the complex plot of the set of complex numbers $c$ for which the sequence of iterates

$f_c (0), f_c (f_c (0)), f_c (f_c (f_c (0))), \dots,$

remains bounded in absolute value.
We say $X$ is locally connected if every point $x\in X$ admits a neighborhood basis consisting entirely of open, connected sets.

Conjecture: The Mandelbrot set $X$ is locally connected.

# Simple unsolved math problem, 5

This is now almost completely solved! Kaisa Matomäki, Maksym Radziwill, Xuancheng Shao, Joni Teräväinen, and Terrance Tao solved the conjecture below in the “interior” of Pascal’s triangle (see T. Tao’s blog post for further details, with the link to the paper and a discussion).

It seems everyone’s heard of Pascal’s triangle. However, if you haven’t then it is an infinite triangle of integers with 1‘s down each side and the inside numbers determined by adding the two numbers above it:

First 6 rows of Pascal’s triangle

The first 6 rows are depicted above. It turns out, these entries are the binomial coefficients that appear when you expand $(x+y)^n$ and group the terms into like powers $x^{n-k}y^k$:

First 6 rows of Pascal’s triangle, as binomial coefficients.

The history of Pascal’s triangle pre-dates Pascal, a French mathematician from the 1600s, and was known to scholars in ancient Persia, China, and India.

Starting in the mid-to-late 1970s, British mathematician David Singmaster was known for his research on the mathematics of the Rubik’s cube. However, in the early 1970’s, Singmaster made the following conjecture [1].

Conjecture: If $N(a)$ denotes the number of times the number $a > 1$ appears in Pascal’s triangle then $N(a) \leq 12$ for all $a>1$.

In fact, there are no known numbers $a>1$ with $N(a)>8$ and the only number greater than one with $N(a)=8$ is a=3003.

References:

[1] Singmaster, D. “Research Problems: How often does an integer occur as a binomial coefficient?”, American Mathematical Monthly, 78(1971) 385–386.

# Simple unsolved math problem, 3

A perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself. For example,  1 + 2 + 3 = 6 implies 6 is a perfect number.

Unsolved Problem: Are there any odd perfect numbers?

The belief, by some, that there are none goes back over 500 years (wikipedia).

If you want to check out some recent research into this problem, see oddperfect.org.

(Another unsolved problem: Are there an infinite number of even perfect numbers?)

# Simple unsolved math problem, 2

In 1911, Otto Toeplitz asked the following question.

Inscribed Square Problem: Does every plane simple closed curve contain all four vertices of some square?

This question, also known as the square peg problem or the Toeplitz’ conjecture, is still unsolved in general. (It is known in lots of special cases.)

Inscribed square, by Claudio Rocchini

Thanks to Mark Meyerson (“Equilateral triangles and continuous curves”,Fundamenta Mathematicae, 1980) and others, the analog for triangles is true. For any triangle T and Jordan curve C, there is a triangle similar to T and inscribed in C. (In particular, the triangle can be equilateral.) The survey page by Mark J. Nielsen has more information on this problem.

Added 2020-07-01: This has apparently been solved by Joshua Greene and Andrew Lobb! See their ArXiV paper (https://arxiv.org/abs/2005.09193).

# Simple unsolved math problem, 1

In 1937 Lothar Collatz proposed the 3n+1 conjecture (known by a long list of aliases), is stated as follows.

First, we define the function $f$ on the set of positive integers:

If the number $n$ is even, divide it by two: $f(n)=n/2$.
If the number $n$ is odd, triple it and add one: $f(n)=3n+1$.

In modular arithmetic notation, define the function $f$ as follows:
$f(n)= {n/2},\ if \ n\equiv 0 \pmod 2$, and $f(n)= {3n+1},\ if \ n\equiv 1 \pmod 2$. Believe it or not, this is the restriction to the positive integers of the complex-valued map $(2+7z-(2+5z)\cos(\pi z))/4$.

The 3n+1 conjecture is: The sequence
$n,\ f(n),\ f^2(n)=f(f(n)),\ f^3(n)=f(f^2(n)),\ \dots$
will eventually reach the number 1, regardless of which positive integer $n$ is chosen initially.

This is still unsolved, though a lot of people have worked on it. For a recent survey of results, see the paper by Chamberland.