Category Archives: elementary

Dividing a square into triangles of equal area

Take a square and divide it down a diagonal, dividing the square into two triangles. Drawing the opposite diagonal now divides it into four triangles. In these two examples, we divided a square into an even number of triangles, all with equal area. Can we divide a square into an odd number of nonoverlapping triangles, […]

Sum of squares a square?

The sum of squares of two integers can be an integer. For example, $3^2 + 4^2 = 25^2$. Those are also the legs of a right-angle triangle. This example is special in that the numbers $3$ and $4$ are consecutive integers! Can you find a sum of squares of three consecutive integers that make a […]

Does this product sequence converge?

Consider the following series: \begin{align*} a_1 &= \frac{4}{3}\\ a_2 &= \frac{4}{3}\frac{9}{8}\\ &\vdots\\ a_n &= \frac{4}{3}\frac{9}{8}\cdots\frac{(n+1)^2}{(n+1)^2-1} \end{align*} In other words, $a_n$ is the product of all the numbers of the form $n^2/(n^2 – 1)$ for $n=2,\dots, n+1$. Does $\lim_{n\to\infty} a_n$ exist?

The Smallest Number Paradox

The smallest number paradox goes like this: consider the natural numbers: 0,1,2,3,… . Each can be specified by a string of characters. For example, "0" itself specifies 0. However, on my computer there are only finitely many bits. Therefore, only finitely many numbers can be specified as a string on my computer. In other words, […]

Fibonacci sequence modulo m

The Fibonacci sequence is an infinite sequence of integers $f_0,f_1,f_2,\dots$ defined by the initial values $f_0 = f_1 = 1$ and the rule $$f_{n+1} = f_n + f_{n-1}$$ In other words, to get the next term you take the sum of the two previous terms. For example, it starts off with: $$1,1,2,3,5,8,13,21,34,55,\dots$$ You can define […]

What is a Liouville number?

An irrational number $r$ is called a Liouville number if for every positive integer $n$ there exists integers $p$ and $q \gt 1$ such that $$| r – p/q | \lt 1/q^n.$$ The requirement that $q \gt 1$ is crucial because otherwise, all irrational numbers would satisfy this definition. Liouville numbers are transcendental. In an […]

All set endomorphisms of a finite field are polynomial

Let $F$ be a finite field. Did you know that given any function $\varphi:F\to F$, there exists a polynomial $p\in F[x]$ such that $\varphi(a) = p(a)$ for all $a\in F$? It's not hard to produce such the required polynomial: $$ p(x) = \sum_{a\in F} \left( \varphi(a)\prod_{b\not= a}(x – b)\prod_{b\not=a}(a-b)^{-1} \prod \right)$$ This works because every […]

Harmonic Numbers

The $n$th harmonic number $h(n)$ is defined as $$h(n) = \sum_{i=1}^n 1/i$$ The harmonic series is the associated series $\sum_{i=1}^\infty 1/i$ and it diverges. There are probably quite a few interesting ways to see this. My favourite is a simple comparison test: $$1/1 + 1/2 + 1/3 + \cdots\\ \geq 1/2 + 1/2 + 1/4 […]

Switching the order of summation

Characteristic functions have magical properties. For example, consider a double summation: $$\sum_{k=1}^M\sum_{r=1}^k a_{r,k}.$$ How do you switch the order of summation here? A geometric way to think of this is arrange the terms out and "see" that this sum must be equal to $$\sum_{r=1}^M\sum_{k=r}^M a_{r,k}.$$ I find this unsatisfactory because the whole point of good […]

When Is Squaring and Cubing a Group Homomorphism?

Let $G$ be a group. Define $\varphi:G\to G$ by $\varphi(x) = x^2$. When is $\varphi$ a homomorphism? Clearly, $\varphi$ is a homomorphism whenever $G$ is abelian. Conversely, if $\varphi$ is a homomorphism then for any $x,y\in G$, we get $xyxy = \varphi(xy) = \varphi(x)\varphi(y) = x^2y^2$. So, $xyxy = xxyy$. Canceling the $x$ from the […]