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The Common Core State Standards Initiative was intended to establish standards for the curriculum for K–12 students in order to universally elevate the the quality of education in the United States. Whether the initiative has succeeded, or not, is a matter of heated debate. In particular, the merits of the mathematics standards are a contentious matter to the extent that colleagues in my math department at UC Berkeley have penned opposing opinions on the pages of the Wall Street Journal (see Frenkel and Wu vs. Ratner). In this post I won’t opine on the merits of the standards, but rather wish to highlight a shortcoming in the almost universal perspective on education that the common core embodies:

The emphasis on what K–12 students ought to learn about what is known has sidelined an important discussion about what they should learn about what is not known.

To make the point, I’ve compiled a list of unsolved problems in mathematics to match the topics covered in the common core. The problems are all well-known to mathematicians, and my only contribution is to select problems that (a) are of interest to research mathematicians (b) provide a good balance among the different areas of mathematics and (c) are understandable by students studying to (the highlighted) Common Core Standards.  For each grade/high school topic, the header is a link to the Common Core Standards. Based on the standards, I have selected a single problem to associate to to the grade/topic (although it is worth noting there were always a large variety to choose from). For each problem, I begin by highlighting the relevant common core curriculum which the problem is related to, followed by a warm up exercise to help introduce students to the problem. The warm ups are exercises that should be solvable by students with knowledge of the Common Core Standards. I then state the unsolved problem, and finally I provide (in brief!) some math background, context and links for those who are interested in digging deeper into the questions.

Disclaimer: it’s possible, though highly unlikely that any of the questions on the list will yield to “elementary” methods accessible to K–12 students. It is quite likely that many of the problems will remain unsolved in our lifetimes. So why bother introducing students to such problems? The point is that the questions reveal a sliver of the vast scope of mathematics, they provide many teachable moments and context for the mathematics that does constitute the common core, and (at least in my opinion) are fun to explore (for kids and adults alike). Perhaps most importantly, the unsolved problems and conjectures reveal that the mathematics taught in K–12 is right at the edge of our knowledge: we are always walking right along the precipice of mystery. This is true for other subjects taught in K–12 as well, and in my view this reality is one of the important lessons children can and should learn in school.

One good thing about the Common Core Standards, is that their structure allows, in principle, for the incorporation of standards for unsolved problems the students ought to know about. Hopefully education policymakers will consider extending the Common Core Standards to include such content. And hopefully educators will not only teach about what is not known, but will also encourage students to ask new questions that don’t have answers. This is because  “there are known knowns; there are things we know we know. We also know there are known unknowns; that is to say we know there are some things we do not know. But there are also unknown unknowns – the ones we don’t know we don’t know.”

Kindergarten

Relevant common core: “describing shapes and space.”

Warm up: can you color the map of Africa with four colors so that no two countries that touch are filled in with the same color?

Can you color in the map with three colors so that no two countries that touch are filled in with the same color?

The unsolved problem: without trying all possibilities, can you tell when a map can be colored in with 3 colors so that no two countries that touch are filled in with the same color?

Background and context: The four color theorem states (informally) that “given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color.” (from wikipedia). The mathematics statement is that any planar graph can be colored with four colors. Thus, the first part of the “warm up” has a solution; in fact the world map can be colored with four colors. The four color theorem is deceivingly simple- it can be explored by a kindergartner, but it turns out to have a lengthy proof. In fact, the proof of the theorem requires extensive case checking by computer. Not every map can be colored with three colors (for an example illustrating why see here). It is therefore natural to ask for a characterization of which maps can be 3-colored. Of course any map can be tested for 3-colorability by trying all possibilities, but a “characterization” would involve criteria that could be tested by an algorithm that is polynomial in the number of countries. The 3-colorability of planar graphs is NP-complete.

Relevant common core: “developing understanding of whole number relationships”.

Warm up: Suppose that in a group of people, any pair of individuals are either strangers or acquaintances. Show that among three people there must be at either at least two pairs of strangers or else at least two pairs of acquaintances.

The unsolved problem: Is it true that among 45 people there must be 5 mutual strangers or 5 mutual acquaintances?

Background and context: This problem is formally known as the question of computing the Ramsey number R(5,5). It is an easier (although probably difficult for first graders) problem to show that R(3,3)=6, that is, that among six people there will be either three mutual strangers or else three mutual acquaintances. It is known that $43 \leq R(5,5) \leq 49$. The difficulty of computing Ramsey numbers was summed up by mathematician Paul Erdös as follows:

“Imagine an alien force, vastly more powerful than us, landing on Earth and demanding the value of R(5, 5) or they will destroy our planet. In that case we should marshal all our computers and all our mathematicians and attempt to find the value. But suppose, instead, that they ask for R(6, 6). In that case we should attempt to destroy the aliens.” – from Ten Lectures on the Probabilistic Method.

Relevant common core: “building fluency with addition and subtraction”.

Warm up: Pascal’s triangle is a triangular array of numbers where each entry in a row is computed by adding up the pair of numbers above it. For example, the first six rows of Pascal’s triangle are:

Write out the next row of Pascal’s triangle.

The unsolved problem: Is there a number (other than 1) that appears 10 times in the (infinite) Pascal’s triangle?

Background and context: The general problem of determining whether numbers can appear with arbitrary multiplicity in Pascal’s triangle is known as Singmaster’s conjecture. It is named after the mathematician David Singmaster who posed the problem in 1971. It is known that the number 3003 appears eight times, but it is not known whether any other number appears eight times, nor, for that matter, whether any other number appears more than eight times.

Relevant common core: “(1) developing understanding of multiplication and division and strategies for multiplication and division within 100”.

Warm up: Practice dividing numbers by 2 and multiplying by 3.

The unsolved problem: Choose a natural number n. If n is even, divide it by 2 to get $n\div 2$. If n is odd, multiply it by 3 and add 1 to obtain $3\times n+1$. Repeat the process. Show that for any initial choice n, the repeating process will eventually reach the number 1.

Background and context: The conjecture is called the Collatz conjecture, after Lothar Collatz who proposed it in 1937. It is deceptively simple, but despite much numeric evidence that it is true, has eluded proof. Mathematician Terence Tao has an interesting blog post explaining why (a) the conjecture is likely to be true and (b) why it is likely out of reach of current techniques known to mathematicians.

Relevant common core: “Determine whether a given whole number in the range 1-100 is prime or composite”.

Warm up: Write the number 100 as the sum of two prime numbers.

The unsolved problem: Show that every even integer greater than 2 can be expressed as the sum of two primes.

Background and context:  This problem is known as the Goldbach conjecture. It was proposed by the mathematician Christian Goldbach in a letter to the mathematician Leonhard Euler in 1742 and has been unsolved since that time (!) In 2013 mathematician Harald Helfgott settled the “weak Goldbach conjecture“, proving that every odd integer greater than 5 is the sum of three primes.

Relevant common core: “Graph points on the coordinate plane”.

Warm up: A set of points in the coordinate plane are in general position if no three of them lie on a line. A quadrilateral is convex if it does not intersect itself and the line between any two points on the boundary lies entirely within the quadrilateral. Show that any set of five points in the plane in general position has a subset of four points that form the vertices of a convex quadrilateral.

The unsolved problem: A polygon is convex  if it does not intersect itself and the line between any two points on the boundary lies entirely within the polygon. Find the smallest number N so that any N points in the coordinate plane in general position contain a subset of 7 points that form the vertices of a convex polygon.

Background and context: The warm up exercise was posed by mathematician Esther Klein in 1933. The question led to the unsolved problem, which remains unsolved in the general case, i.e. how many points are required so that no matter how they are placed (in general position) in the plane there is a subset that form the vertices of a convex n-gon. There are periodic improvements in the upper bound (the most recent one posted on September 10th 2015), but the best current upper bound is still far from the conjectured answer.

A set of 8 points in the plane containing no convex pentagon.

Relevant common core: “Represent three-dimensional figures using nets made up of rectangles and triangles”.

Warm up

The dodecahedron is an example of a convex polyhedron. A convex polyhedron is a polyhedron that does not intersect itself and that has the property that any line joining two points on the surface lies entirely within the polyhedron. Cut out the net of the dodecahedron (shown above) and fold it into a dodecahedron.

The unsolved problem: Does every convex polyhedron have at least one net?

Background and context: Albrecht Dürer was an artist and mathematician of the German Renaissance. The unsolved problem above is often attributed to him, and is known as Dürer’s unfolding problem. It was formally posed by the mathematician Geoffrey Shephard in 1975.

One of my favorite math art pieces: Albrecht Dürer’s engraving Melencolia I.

Relevant common core: “Analyze proportional relationships and use them to solve real-world and mathematical problems.”

Warm up: Two runners, running at two different speeds $v_0$ and $v_1$, run along a circular track of unit length. A runner is lonely at time t if she is at a distance of at least 1/2 from the other runner at time t. If both runners all start at the same place at t=0, show that the runners will both be lonely at time $t=\frac{1}{2(v_1-v_0)}$.

The unsolved problem: Eight runners, each running at a speed different from that of the others, run along a circular track of unit length. A runner is lonely at time t if she is at a distance of at least 1/8 from every other runner at time t. If the runners all start at the same place at t=0, show that each of the eight runners will be lonely at some time.

Background and context: This problem is known as the lonely runner conjecture and was proposed by mathematician J.M Wills in 1967. It has been proved for up to seven runners, albeit with lengthy arguments that involve lots of case checking. It is currently unsolved for eight or more runners.

Relevant common core: “Know and apply the properties of integer exponents”.

Warm up: Show that $3^{3n}+[2(3^n)]^3 = 3^{3n+2}$ for any integer greater than or equal to 1.

The unsolved problem: If $A^x+B^y=C^z$ where A,B,C,x,y and are positive integers with $x,y,z>2$ then A,B and C have a common prime factor.

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