I'm looking for some mathematics that will challenge me as a year $12$ student.

Question

I am an upcoming year $12$ student, school holidays are coming up in a few days and I've realised I'm probably going to be extremely bored. So I'm looking for some suggestions.

I want a challenge, some mathematics that I can attempt to learn/master. Obviously nothing impossible, but mathematics is my number $1$ favorite thing and I really want something to keep me busy and something that can further my understanding of mathematics. Also I would be interested in any mathematical focused book suggestions.

So far in school I've done the usual:

Matrices, transformation matrices, Sine Cosine and Tangent (graphs and proofs), lots and lots of parabolas/quadratics, statistics, growth and decay, calculus intro, Calculus derivation and integration, vectors, proof by induction and complex numbers.

Any suggestions would be heavily appreciated.

Answer 1

I have been asked for a hint. Gauss was thinking about roots of unity. The method of Gauss for dealing with these polynomials is in chapter 9 of Cox Galois Theory. In fact, as the author points out, this work predates Galois Theory by about thirty years. To give a name, although it will not help with these problems, https://en.wikipedia.org/wiki/Gaussian_period

---

For the indicated prime in the simplest problem below, define $$ \omega = e^{2 \pi i / p} = \cos \frac{2 \pi }{p} + i \sin \frac{2 \pi }{p}$$ Then $$ \omega^p = 1 $$ and the indicated real root, when a single cosine term, is just $$ \omega + \frac{1}{\omega} = \omega + \omega^{p-1}$$

For the problems with two cosine terms, the indicated root is, with some integer $k,$ $$ \omega + \omega^k + \omega^{p-k} + \omega^{p-1}$$

---

Show that $$ x = 2 \cos \left( \frac{2 \pi}{7} \right) $$ is a root of $$ x^3 + x^2 - 2 x - 1. $$ For this one, find all the roots. This appears on page 6 of Reuschle

---

Show that $$ x = 2 \cos \left( \frac{2 \pi}{11} \right) $$ is a root of $$ x^5 + x^4 -4 x^3 -3 x^2 + 3 x + 1. $$ This appears on page 9 of Reuschle

---

Show that $$ x = 2 \cos \left( \frac{2 \pi}{23} \right) $$ is a root of $$ x^{11} + x^{10} - 10 x^9 - 9 x^8 + 36 x^7 + 28 x^6 - 56 x^5 - 35 x^4 + 35 x^3 + 15 x^2 - 6 x - 1. $$ This one appears on page 30 of Reuschle

---

Show that $$ x = 2 \cos \left( \frac{2 \pi}{47} \right) $$ is a root of $$ x^{23} + x^{22} - 22 x^{21} - 21 x^{20} + 210 x^{19} + 190 x^{18} -1140 x^{17} -969 x^{16} + 3876 x^{15} + 3060 x^{14} $$ $$ -8568 x^{13} - 6188 x^{12} + 12376 x^{11} + 8008 x^{10} - 11440 x^9 - 6435 x^8 + 6435 x^7 + 3003 x^6 - 2002 x^5 $$ $$ -715 x^4 + 286 x^3 + 66 x^2 - 12 x - 1 $$ This one appears on page 73 of Reuschle. Really nice. I have ordered a cheap paperback reprint of Reuschle.

Note that $$ 2, 5, 11, 23, 47 $$ are the maximal chain of Sophie Germain primes (well, at least when they are not $4 \pmod 5$); apparently $47$ is called a "safe prime" instead. In any string of integers $x_1,x_2,x_3,x_4,x_5,$ such that $x_{n+1} = 2 x_n + 1,$ one of the string is divisible by $5.$ The five numbers can only be primes if one of them is equal to $5.$ Meanwhile, $ x = 2 \cos \left( \frac{2 \pi}{47} \right) $ is a root of $x^2 + x - 1,$ which begins the chain of Gaussian minimal polynomals.

We can get a longer chain when the first one is $-1 \pmod {30},$ i.e. $$ 89, \; 179, \; 359, \; 719, \; 1439, \; 2879. $$ $$ 1122659, \; 2245319, \; 4490639, \; 8981279, \; 17962559, \; 35925119, \; 71850239. $$

---

Find at least one root of $$ x^3 + x^2 - 4 x + 1. $$ Sum of a pair of cosines this time, denominator $13$. It is actually quite unusual to have one of these where the "main" root is a single cosine term. That happens only when the degree is prime $q$, while $2q+1$ is also prime, while the polynomial is constructed very carefully; the recipe is due to Gauss. Section seven in the Disquisitiones Arithmeticae.

---

Find at least one root of $$ x^7 + x^6 - 12 x^5 - 7 x^4 + 28 x^3 + 14 x^2 - 9 x + 1. $$ Sum of a pair of cosines this time, denominator $29$. This one appears on page 35 of Reuschle

Answer 2

Project Euler is a great source of interesting problems. Many of them require you to learn a little computer programming, which I highly recommend you try if you haven't before. (And if you don't have a preferred programming language, give Sage a try. Nice clean syntax with an extensive math library and you don't even have to install anything.)

---

Just for a sense of flavor, here's an early problem about the Collatz Conjecture that I rather like. It's slow by brute force, but a bit of recursive magic solves it in under a second.

The following iterative sequence is defined for the set of positive integers:

$n \rightarrow n/2$ ($n$ is even)

$n \rightarrow 3n + 1$ ($n$ is odd)

Using the rule above and starting with $13$, we generate the following sequence:

$13 \rightarrow 40 \rightarrow 20 \rightarrow 10 \rightarrow 5 \rightarrow 16 \rightarrow 8 \rightarrow 4 \rightarrow 2 \rightarrow 1$

It can be seen that this sequence (starting at $13$ and finishing at $1$) contains $10$ terms. Although it has not been proved yet (Collatz Problem), it is thought that all starting numbers finish at $1$.

Which starting number, under one million, produces the longest chain?

Answer 3

If you are really seeking for recomandation I can suggest you some books:

$1$.Putnam and beyond: Book by Razvan Gelca and Titu Andreescu.

$2$. Elementary Number Theory: Primes, Congruences, and Secrets:By Willaim Stein.

$3$.Mathematical Diamonds:By Ross Honsberger.

These three books are the best collection for developing strong logical skills and mastering the problem solving abilities. Other's opinion can be different from that of mine but since I m also an upcoming year $12$ student so I thought you will like what I like.

Answer 4

There are a lot of good answers already to this question, but I thought I could contribute with some of my favourite "elementary" problems (ones which barely require any prerequisites). In my opinion all of these problems have beautiful solutions which are a joy to find, and altough they don't necessarily lead to deep mathematics, they do point you towards some general problem solving strategies (some of which you may have not yet encountered in school).

1. Show that given $7$ points in the unit circle there will always be two which have distance less than $1$.

2. On an island there are $13$ grey, $15$ brown and $17$ green chameleons. Whenever two differently colored chameleons meet they frighten each other so much that both of them turn into the third color. Is it possible that after some time all chameleons will be of the same color?

3. Show that the sum of the the reciprocals of $100$ odd numbers can never be equal to $1$.

4. You have a rectangular array of real numbers. One step consists of multiplying each element of a column or a row by $-1$. Can you always reach a state, in a finite number of steps, where the sum of the elements in each row and column are non-negative?

5. Two opposing corners of a chess board are removed. Can you cover the rest of the board with dominoes of size $2 \times 1$? What if only one corner is removed and the dominoes are of size $3 \times 1$? (This one is a well known and beautiful problem.)

Answer 5

1) Establish a formula for the sum of the first $n$ integers. Then for the sum of the first $n$ perfect squares ($n^2$). Then cubes... Go as far as you can.

2) Given $n$ points $(x_i,y_i)$, find a polynomial such that $p(x_i)=y_i$ for all $i$. Try with $n=1$, then $n=2$, then $3$... and try to find a general approach. Also try to minimize the amount of computation.

3) Find the twentieth derivative of $\tan(x)$.

4) Compute $e$ by hand to the $25^{th}$ decimal.

5) Compute $\pi$ by hand to the $25^{th}$ decimal.

6) Draw a random curve and find an equation that matches it.

7) Find a way to compute the factorial of a decimal number, like $5.3!$

8) Learn about the Bernouilli, Euler, and Stirling numbers.

Answer 6

This question probably has as many answers as there are mathematicians. I'll plug the following list of books that make a great introduction to a field:

Finite-Dimensional Vector Spaces, Halmos

Fourier Analysis, Stein & Shakarchi

Real Mathematical Analysis, Pugh

Munkres, Topology

Artin, Abstract Algebra

Topology From The Differentiable Viewpoint

All of these books provide great introductions to a very cool part of mathematics and are very hard. My introduction to mathematics was picking up the third book on this list and banging my head against it until I got through it. Good luck.

Answer 7

Notice to the reader: at the end on list item #8

---

Start by proving this 'simple' sum:

$$\frac{1-r^{n+1}}{1-r}=1+r+r^2+r^3+\dots+r^n$$

After you've proven that, differentiate it and solve the following sum

$$1+2+3+\dots+n=\lim_{r\to1}\dots$$

See if you can derive what higher degrees are:

$$1^2+2^2+3^2+\dots+n^2=???$$

Aside from that, do the opposite: integrate! (choose your bounds wisely)

$$1+\frac12+\frac13+\dots+\frac1n=?$$

Let $n\to\infty$ to reveal what happens to the famous sum $1+\frac12+\frac13+\dots$

On the contrary, derive the integral that solves the following, then let $n\to\infty$:

$$1-\frac12+\frac13-\frac14+\dots+(-1)^{n+1}\frac1n=?$$

See if you can figure out the formula for

$$r^x-r^{x+1}+r^{x+2}-r^{x+3}+\dots+(-1)^nr^{x+n}$$

Use it to derive what the following infinite sum is:

You may need to use the fact $\arctan(x)=\int\frac1{1+x^2}dx$ and some $u$ substitution.

$$\frac1{0.5}-\frac1{1.5}+\frac1{2.5}-\frac1{3.5}+\dots=$$

Taking a side turn, consider the following function:

$$y=\frac{\cos(x)+i\sin(x)}{e^{ix}}$$

what's its derivative? Should tell you something really deep about the way complex numbers work.

Using this information, and the very first sum we started with, can you now calculate what this sum is?

$$\cos(1)+\cos(2)+\dots+\cos(n)$$

How about for $\sin$?

Now lets get crazy. Can you calculate what this sum is???

$$\frac{\sin(1)}1+\frac{\sin(2)}2+\frac{\sin(3)}3+\dots$$

---

Now that was all fun, but here are some even more fun (and possibly very useful) functions/theorems you might want to poke around with.

$0!$| The Gamma function (remember the factorial?)

$\Gamma(3)$| The Riemann Zeta function $\displaystyle\sum_{k=1}^\infty\frac1{k^s}$ (no, you can't use the above methods here)

$18\zeta(2)/\pi^2$| The Dirichlet eta function $\displaystyle\sum_{k=1}^\infty\frac{(-1)^{k+1}}{k^s}$

4.0| Taylor series/Taylor's theorem (derive binomial expansion with this)

4.1| Laurent series (probably too advanced, this is complex analysis material)

$e^{\ln(5)}$| This guy named "Leonhard Euler" (you really should read up on him)

$e^{\pi}-17.14$| Analytic function (and its properties)

$7^{13}\mod13$| MSE $\leftarrow$ fundamental theorem of my free time

$2^3$| ^ See the above and use it whenever you need.

Answer 8

Consider learning the use of the Burnside lemma, the symbolic factorization of permutations into cycles and the computation of cycle indices of common permutation groups (e.g. cyclic, dihedral) and how to apply them to edge / face / vertex colorings of polyhedra, then move on to the Polya Enumeration Theorem. The text I would suggest for this is Harary and Palmer, Graphical Enumeration. Basically requires no pre-requisites. Choose a CAS to learn if you haven't already.

Answer 9

These are some integral-themed questions I had fun with at your age.

Evaluate

$$\int_0^{\pi}\log\sin x\,dx$$

$$\int_0^{\pi/2}\frac{\sin^a(x)}{\sin^a(x)+\cos^a(x)} \,dx\\\vphantom{\cfrac11}$$

Show that:

for $n\in \mathbb Z^{\ge0}$

$$n! = \int_0^{\infty}x^ne^{-x} \,dx$$

and with this result show that

$$\int_0^1 x^{-x}\,dx=\sum_{n=1}^\infty n^{-n}$$

---

Show that the limit exists:

$$\lim_{n\to\infty}1+\frac{1}{2}+\cdots+\frac{1}{n}-\log n$$

Answer 10

Well my suggestion would be to learn about (in this particular order):

1. Infinite series (converging and diverging tests).

2. Taylor form (Taylor-maclaurin approximation function).

3. Evaluation of different equations and the solution of these equations with Taylor forms.

After wards, if you see that it really interests you, you can continue to approximation theorem or learn about multi variable functions instead (only if you pretty sure about your calculus skills (at least calculus 1)).

Well I hope that will give you some ideas, have fun!

Answer 11

When I was exactly your age I spent the summer working out how to solve cubic equations:

$$ ax^3 + bx^2 + cx +d = 0 $$

Answer 12

I was at a similar level of experience when I reached U.S. 12th grade. (I'm not sure what "Year 12" refers to.)

What I eventually found was that within the branches of math you have mentioned, you have had virtually zero experience with discrete mathematics.

The mathematics you have done deal with numbers that are infinitely divisible (real numbers), which is appropriate for dealing with physics quantities such as distance, speed, force, mass, acceleration, etc. These mathematics are all continuous mathematics.

However, there is an entirely different sort of mathematics, which deals with separate quantities—such as "true" or "false." "Spades" or "Clubs" or "Diamonds." There is nothing in between.

At first glance it would seem there is little of interest in studying only integers—after all, they are a subset of the real numbers—but that is the subject called Number Theory and it is the foundation for all modern cryptography!

Other branches of discrete mathematics include Combinatorics (counting), and Graph Theory.

I highly recommend the materials from MIT Open CourseWare. The particular course I started with was "Mathematics for Computer Science."

For a bright high school student with the math you've studied under your belt, you should have no major trouble with the material—but plenty of interesting new ideas.

Happy learning. :)

Answer 13

(Note: Originally posted as comment, writing as an answer upon request.)

Seek to understand!

Every single result you have learned to apply, now seek to understand that result from the ground up so that you see it with the clarity of 1+1=2.

Differential and integral calculus. Matrices and linear algebra. Trigonometry. Complex numbers. Intelligently asking (yourself then if necessary others) well-directed questions is key. Just by formulating the question, the brain is put into an optimal state for receiving the answer.

The primary question is not what you should be learning but how you should be learning.

I would not advise anyone to study pure mathematics at University. All the resources are here!

• Wikipedia
• This site (+chatroom)
• ##math on IRC Freenode
• Coursera
• Youtube
• etc.

You can train yourself up to graduate level just from these resources. Go for it! Don't wait for the (broken) educational system -- forge your own path.

From the comfort of your own home you can watch Leonard Susskind explain General Relativity! If you hunt around on IRC you can find theoretical physicists able to explain it to you.

If you are mathematically inclined, now is a great time to be alive!

Answer 14

It depends on what catches your eye. Having fun is important! At your point in mathematics education, you have a variety of options available for self-study. Here's a possible list of topics, and your future path from then on:

• Multivariable calculus (followed by vector calculus, complex analysis, and Fourier analysis)
• Ordinary differential equations (followed by partial differential equations)
• Abstract algebra
• Discrete math (followed by graph theory)
• Logic
• Number theory
• Numerical analysis
• Python (followed by useful tools such as matplotlib and SymPy, or endeavors into scientific computing)

Other topics to look forward to once you've become comfortable in undergraduate mathematics include topology, measure theory, differential geometry, and fluids.

If you want a taste of applications, you may consider a topic which intrigues you, such as machine learning or neural networks. (These only require knowledge of basic statistics.)

Also, it may not hurt to do a more rigorous study of topics you already "know". For instance, consider topics covered in 'advanced' linear algebra classes. Or perhaps try applying what you know to solve problems, i.e. Project Euler.

Answer 15

I applaud your mathematical enthusiasm! My preference once I started my maths degree was for applied things, so the maths behind the weather! Try that as a research starting point and see where it takes you.

Else I second the recommendation for programming, I got taught c++ on my maths degree, and getting computers to do maths for you is really important in lots of areas.

Have fun!

Answer 16

Solving problems is fun.

It used to be said that mathematics and cricket were not spectator sports; and this is still true of mathematics. To progress as a mathematician, you have to strengthen your mathematical muscles. It is not enough just to read books or attend lectures. You have to work on problems yourself.

If you're considering studying maths at university, you might like to try the problems in Stephen Siklos' Advanced Problems in Mathematics. These assume no more knowledge than of UK A-level syllabus, but are long university-style questions that get you to think, proving some interesting results in the process. (I'd go so far as to say these problems are generally better written and more engaging than those in my university exams.) To quote the introduction:

The general aim is to help bridge the gap between school and university mathematics. You might wonder why such a gap exists. The reason is that mathematics is taught at school for various purposes: to improve numeracy; to hone problem-solving skills; as a service for students going on to study subjects that require some mathematical skills (economics, biology, engineering, chemistry — the list is long); and, finally, to provide a foundation for the small number of students who will continue to a specialist mathematics degree. It is a very rare school that can achieve all this, and almost inevitably the course is least successful for its smallest constituency, the real mathematicians.

The questions are taken from past STEP papers, which are the entrance exams for the Cambridge Mathematical Tripos. Don't be alarmed if you can't solve any at first. This happened to me at sixteen. I'd always found maths at school easy, so it came as quite a shock to my ego. Working through the problems teaches the kind of thinking demanded by mathematics at university, so you'll be well prepared if you choose to apply.

You can buy the book in paperback, or download a PDF from the author for free.

Answer 17

You could look at some of the Questions tagged recreational-mathematics, soft-question and big-list (sort them by "votes").