bonsoon's page
https://radiofreemath.org
Math writings, puzzles, and notes2023-05-20bonsoon# Polynomial nonhomogeneous terms in a linear recurrence.
https://radiofreemath.org/read/posts/0+inbox/Nonhomogeneous-linear-recurrences.md
2023-05-20 12:18Suppose we have some sequence $(a_{n})$ over reals or complex such that it satisfies some $k$-th order linear recurrence $$
a_{n+k}+c_{1}a_{n+k-1}+\cdots+c_{k}a_{n}=p(n)
$$....(read more)# Solution space to linear recurrences.
https://radiofreemath.org/read/posts/0+inbox/solution-space-to-linear-recurrences.md
2023-05-20 12:10Given an $k$-th order linear homogeneous linear recurrence $$
a_{n+k}+c_{1}a_{n+k-1}+\cdots+c_{k}a_{n}=0
$$we seek all sequences $(a_{n})$ over field $\mathbb{C}$, say that solves it. We claim the space is $k$-dimensional. ....(read more)# About.
https://radiofreemath.org/read/posts/About.md
2023-05-17 12:41Just a collection of puzzles, writings, mathematical folklores, and things I find interesting. I will try to update some notes, problems, and quizzes for students and teachers as well.
You can click on [[random]] to browse a random article. ![[---images/---assets/---icons/exclaim-icon.svg]] RSS feed here.
Radio free math, broadcasting math to the world, one shape and number at a time! ....(read more)# Celestial navigation - Part 2 North Star and the Southern Cross.
https://radiofreemath.org/read/posts/summer+program+2023/week+1/Celestial_navigation_2.md
2023-05-16 22:30
You may have seen long exposure images of the night sky like this:
![[summer program 2023/week 1/---files/Celestial_navigation 2023-05-16 11.58.44.excalidraw.svg]]....(read more)# Summer Program 2023 Draft
https://radiofreemath.org/read/posts/summer+program+2023/Summer_program_2023.md
2023-05-16 14:35The first three weeks will emphasize on geometry, trigonometry, vectors, algebra and polynomials, complex numbers, and the last three week will emphasize on calculus, its applications, basic probability and counting. However, related ideas will cross over to give a more comprehensive view of our mathematical tools. We will draw many examples from the natural sciences. Every week there is also a computer lab component where we would learn how to use Python and spreadsheet programs to solve problems.
|Monday|Tuesday|Wednesday|Thursday|Friday|....(read more)# Celestial navigation - Part 1 Latitudes.
https://radiofreemath.org/read/posts/summer+program+2023/week+1/Celestial_navigation.md
2023-05-16 13:03Let us pose the question, how do we find our position on Earth? The ancients of the past and the trailblazers used stars to help guild them, long before GPS were sent into orbit. Let us see how this works!
To start, one would place some kind of imaginary coordinate system on Earth, and indeed we do, they are the latitudes and longitudes (which is essentially **spherical coordinates** with a fixed radius, that one encounters in multivariate calculus!). We shall define them and begin with **latitudes**. We will assume that the Earth is a **perfect sphere** (which it is not), but it will be a good start. ....(read more)# Celestial navigation :
https://radiofreemath.org/read/posts/summer+program+2023/week+1/Celestial_navigation_3.md
2023-05-16 13:03rotation and sunrise
## The rotation.....(read more)# Self-assessment 1
https://radiofreemath.org/read/posts/smc-summer-2023-math-8/quiz-self-assessment-1.md
2023-05-16 10:32What is $\pi$ rounded to 2 decimal places?
[== 3.14]
....(read more)# Math 8 Calculus 2
https://radiofreemath.org/read/posts/smc-summer-2023-math-8/smc-summer-2023-math-8-course-page.md
2023-05-15 15:43**Welcome to Math 8 Calculus 2 !**
![[---images/---assets/---icons/important-icon.svg]] **Important links**....(read more)# s-test-quiz-v1
https://radiofreemath.org/read/posts/quiz/test+quiz+1.md
2023-05-15 15:14What is $\pi$ rounded to 2 decimal places?
[== 3.14]
....(read more)# Math 8 Calculus 2
https://radiofreemath.org/read/posts/smc-summer-2023-math-8/smc-summer-2023-math-8-syllabus.md
2023-05-15 10:30**Instructor:** Bon-Soon LinΒ
**Textbook:** Calculus (Stewart, Clegg, Watson, 9E)
**Lecture times and location:** 06:30PM-09:15PM MTWH at MC 66....(read more)# Teaching
https://radiofreemath.org/read/posts/Teaching.md
2023-05-15 10:24## 2023
- [[smc-summer-2023-math-8/smc-summer-2023-math-8-course-page|Math 8 : Calculus 2]]
- Real analysis and linear algebra review....(read more)# Japanese theorem of cyclic polygons.
https://radiofreemath.org/read/posts/0+inbox/Japanese+theorem+of+cyclic+polygons.md
2023-05-15 09:28Take a convex **cyclic polygon**, that is, a convex polygon whose vertices all lie on a circle. Now take any two triangulations, and draw in all the incircles for each small triangle in the triangulations, like so:
![[0 inbox/---files/---Japanese theorem of cyclic polygons 2023-05-14 10.44.27.excalidraw.svg]]
%%[[0 inbox/---files/---Japanese theorem of cyclic polygons 2023-05-14 10.44.27.excalidraw.md|π Edit in Excalidraw]], and the [[0 inbox/---files/---Japanese theorem of cyclic polygons 2023-05-14 10.44.27.excalidraw.dark.svg|dark exported image]]%%....(read more)# Random conflict free knights.
https://radiofreemath.org/read/posts/0+inbox/Random+conflict+free+knights.md
2023-05-15 09:28Consider a $3\times 3$ chess board, where each square has $p=1/2$ probability of having a knight or not.
What is the probability that this board is free of conflict, that is to say, no knight attacking another knight? ....(read more)# Switch the knights.
https://radiofreemath.org/read/posts/0+inbox/Switch+the+knights.md
2023-05-15 09:28Warm up. A usual knight moves in a 2-by-1 $L$ shape. Consider two knights of one color (say black)) and two knights of another color (say white) are placed in a $3\times 3$ chess board below. Can you move them so the positions of the black knights and the white knights are switched? Note, knights shouldn't move into a square that is occupied. You don't have to move the pieces in alternating colors like in real chess, but try if possible.
![[---images/Switch the knights 2023-05-04 10.15.39.excalidraw.svg]]....(read more)# Three-page embedding of graphs and knots.
https://radiofreemath.org/read/posts/0+inbox/Three+page+embedding+of+graphs+and+knots.md
2023-05-15 09:27(I'll have to revisit this)
A classic question to ask about a graph is whether it can be drawn (embedded) on a plane without intersecting itself. Such a graph is called **planar graph**. As we know, not all graphs are planar, and by Kuratowski theorem we know that a graph is planar if and only if it contains a subgraph that is a copy of $K_5$ or $K_{3,3}$ (or some subdivision of them). A natural generalization of planarity is to ask can a graph be embededded (where edges don't intersect) in a **book** of $k$-many pages, which is $k$ many half-planes all glued at their common boundary, forming the **seam** or spine of the book. In this $k$-page book world, the seam is where one can travel between the pages. (Note each page has both sides the same, there is no double sided-ness.)![[---images/Three page embedding of graphs and knots 2023-05-05 15.44.05.excalidraw.svg]]%%[[---images/Three page embedding of graphs and knots 2023-05-05 15.44.05.excalidraw|π Edit in Excalidraw]], and the [[Excalidraw/Three page embedding of graphs and knots 2023-05-05 15.44.05.excalidraw.dark.svg|dark exported image]]%%
So the natural question to ask is: When can a graph be embedded in some $k$-page book, for some $k$? As it turns out, **every graph can be embedded in a $3$-page book !** And similarly, **every knot and link can be embedded in a $3$-page book** as well! (At least, when the graph and knot are "finite" enough.)....(read more)# Ptolemy theorem with complex numbers.
https://radiofreemath.org/read/posts/0+inbox/Ptolemy-theorem-with-complex-numbers.md
2023-05-15 09:27Ptolemy theorem states that given a cyclic quadrilateral (a quadrilateral inscribed in a circle), the product of the diagonals is equal to the sum of the product of the lengths of the opposing sides:
![[0 inbox/---files/Ptolemy-theorem-with-complex-numbers 2023-05-14 12.13.26.excalidraw.svg]]
%%[[0 inbox/---files/Ptolemy-theorem-with-complex-numbers 2023-05-14 12.13.26.excalidraw.md|π Edit in Excalidraw]], and the [[0 inbox/---files/Ptolemy-theorem-with-complex-numbers 2023-05-14 12.13.26.excalidraw.dark.svg|dark exported image]]%%....(read more)# Hero's formula - a complex numbers proof.
https://radiofreemath.org/read/posts/0+inbox/Heros-formula-complex-numbers.md
2023-05-15 09:27Given an arbitrary triangle $T$ with sides $a,b,c$, Hero's formula states its area $A$ is given by
$$
A = \sqrt{s(s-a)(s-b)(s-c)}....(read more)# Flip graphs of convex polygons.
https://radiofreemath.org/read/posts/0+inbox/Flip-graphs-of-convex-polygons.md
2023-05-15 09:26Consider a convex polygon $P$ in the plane, and consider all its triangulations. For an $n$-gon, it is famously known that the number of triangulations is given by the **Catalan numbers** $C_{n-2}$, where $C_n = \frac{1}{n+1}{2n \choose n}$. For instance, here are the $\frac{1}{5}{8 \choose 4} = 14$ triangulations of a convex hexagon:
![[0 inbox/---files/Flip-graphs-of-convex-polygons 2023-05-14 08.52.05.excalidraw.svg]]
%%[[0 inbox/---files/Flip-graphs-of-convex-polygons 2023-05-14 08.52.05.excalidraw|π Edit in Excalidraw]], and the [[---files/Flip-graphs-of-convex-polygons 2023-05-14 08.52.05.excalidraw.dark.svg|dark exported image]]%%....(read more)# Cosmic distance ladder : Size of the Earth
https://radiofreemath.org/read/posts/summer+program+2023/week+1/Cosmic_distance_ladder.md
2023-05-13 20:40Is it possible to measure large distances without actually measuring large dsitances? A famous story goes that the ancient Greek **Eratosthenes** was able to estimate the size of the Earth without, of course, actually measure out the entire Earth. Let's see how they did this!
===....(read more)# Cosmic distance ladder : The size of the Sun!
https://radiofreemath.org/read/posts/summer+program+2023/week+1/Cosmic_distance_ladder_5.md
2023-05-12 09:18How about ... the sun. Can we estimate its size ?
As it turns out, during a solar eclipse the moon blocks out the sun almost completely, where the moon overlaps the sun nearly perfectly. Here is an image of the 1999 solar eclipse, taken from Wikipedia:....(read more)# Python exercise 1.
https://radiofreemath.org/read/posts/summer+program+2023/week+1/Python_week_1.md
2023-05-12 01:35![[---images/---assets/---icons/computer-icon.svg]] Write a Python function called ```size_of_sun``` to estimate of the size of the sun using the distance from Earth to the Sun, the distance from Earth to the moon, and the size of the moon. Your function should look something like this
```python
def size_of_sun(dist_to_sun, dist_to_moon, size_of_moon):....(read more)# Cosmic distance ladder : To the Moon!
https://radiofreemath.org/read/posts/summer+program+2023/week+1/Cosmic_distance_ladder_2.md
2023-05-11 16:47The ancient Greeks were also able to find the distance to the moon, again of course, without physically measuring it. By now they knew the size of the Earth, and using some clever observation about the moon and **lunar eclipses**, they were able to estimate the distance to the moon !
Let us see how this might be done.....(read more)# Cosmic distance ladder : To the Sun!
https://radiofreemath.org/read/posts/summer+program+2023/week+1/Cosmic_distance_ladder_3.md
2023-05-11 15:46What about the sun ? Can we find out how far it is ? Aristarchus famously wrote about how one could find this distance, using a realization about **moon phases** !
## Half-moon phase.....(read more)# Cosmic distance ladder : The size of the moon.
https://radiofreemath.org/read/posts/summer+program+2023/week+1/Cosmic_distance_ladder_4.md
2023-05-11 13:15Ok let us keep going, how about the size of the moon? To do this we will use the **curvature of Earth's shadow** on the moon !
Imagine the situation when the moon enters and leaves Earth's shadow during an eclipse. Let use try to find how big that shadow is in comparison to the moon. When viewed from the Earth to the moon, the moon moves into a circular shadow casted by the Earth:....(read more)# Trigonometry, law of cosine and the law of sine
https://radiofreemath.org/read/posts/summer+program+2023/week+1/Trigonometry_cosine_sine.md
2023-05-11 11:51Let us review some basic trigonometric definitions by the following diagram, recall one way to define them is that they are **ratios of lengths** in a right triangle:
![[summer program 2023/week 1/---files/Trigonometry_cosine_sine 2023-05-10 08.09.19.excalidraw.svg]]
%%[[summer program 2023/week 1/---files/Trigonometry_cosine_sine 2023-05-10 08.09.19.excalidraw.md|π Edit in Excalidraw]], and the [[summer program 2023/week 1/---files/Trigonometry_cosine_sine 2023-05-10 08.09.19.excalidraw.dark.svg|dark exported image]]%%....(read more)# Pythagorean theorem and orthogonality
https://radiofreemath.org/read/posts/summer+program+2023/week+1/Pythagorean_theorem_and_orthogonality.md
2023-05-11 11:51One of the first geometric fact that you may have learned is the Pythagorean theorem, which states:
> **Theorem.** Pythagorean theorem.....(read more)# When is a group a union of 3 proper subgroups?
https://radiofreemath.org/read/posts/0+inbox/When+is+a+group+a+union+of+3+proper+subgroups.md
2023-05-11 11:51Let $G$ be a group. Then $G$ is a union of three proper subgroups if and only if there exists some normal subgroup $H$ such that $G/H\approx V_4$, the Klein-4 group.
---....(read more)# What the shape: A dangling rope.
https://radiofreemath.org/read/posts/0+inbox/What+the+shape+dangling+rope.md
2023-05-11 11:51Take a piece of rope or necklace, and lift the two ends up and let the rope dangle naturally by gravity. What is the shape made by this rope? The shape *looks parabolic*, but is it really a parabola? Call this shape the graph of some function $y = f(x)$. We would like to find out what $f(x)$ is.
![[---images/What the shape - dangling chain 2023-05-05 11.39.32.excalidraw.svg]]%%[[---images/What the shape - dangling chain 2023-05-05 11.39.32.excalidraw|π Edit in Excalidraw]], and the [[Excalidraw/--- What the shape - hanging chain 2023-05-05 11.39.32.excalidraw.dark.svg|dark exported image]]%%
If you like to know the answer now, you can jump to the end. But let us see if we can derive this shape. Of course, we will need to make some **assumptions** and some simple physics, that everything is idealized in the following:....(read more)# Reassembling a cube.
https://radiofreemath.org/read/posts/0+inbox/Reassembling+a+cube.md
2023-05-11 11:51Consider a $3\times 3\times 3$ cube made of 27 smaller $1\times 1\times 1$ cubes. The outer 6 faces of the $3\times 3\times 3$ cube is painted black.
Now shuffle all the tiny cubes and assemble them back randomly into a $3\times 3\times 3$ cube again. What is the probability that the surface is all black again?....(read more)# Rolling a coin around other coins.
https://radiofreemath.org/read/posts/0+inbox/Rolling+a+coin+around+other+coins.md
2023-05-11 11:51Take two coins of the same size and place them touching each other on the table. Now fixing one of them, roll the other coin without slipping around the fixed coin. As you make the rolling coin roll around the fixed coin once, how many times has the rolling coin turn around? Is it, (1) One time, (2) two times, (3) three times, (4) other?
![[Excalidraw/Rolling a coin around other coins 2023-05-07 18.41.10.excalidraw.svg]]And what if the fixed coin is **twice** as large as the rolling coin? Or if the rolling coin rolls over **several** fixed coins?
![[Excalidraw/Rolling a coin around other coins 2023-05-07 19.00.33.excalidraw.svg]]....(read more)# Pairing points with disjoint line segments.
https://radiofreemath.org/read/posts/0+inbox/Pairing+with+disjoint+line+segments.md
2023-05-11 11:51Given $2n$ points in the plane in **general position**, that is, **no three are colinear**. Color $n$ of them blue and $n$ of them red. Is it always possible to pair off a blue and a red point with a **straight line segment** in such a way that the line segments **never intersect**?
![[---images/--- pairing with disjoint line segments 2023-05-06 15.41.37.excalidraw.svg]]
If so, why? If not, can you come up with a configuration of points so no matter you pair them, there will be two line segments that intersect? ....(read more)# Maximal rooks and bishops.
https://radiofreemath.org/read/posts/0+inbox/Maximal+rooks+and+bishops.md
2023-05-11 11:51Given an $n \times m$ chessboard $B$, it is not hard to see that there can be at most $\min(n,m)$ many rooks that you can place on $B$ where the rooks are all non-attacking each other.
---....(read more)# Godly magic squares.
https://radiofreemath.org/read/posts/0+inbox/Godly+magic+squares.md
2023-05-11 11:51A classical magic square of order $n$ is an $n\times n$ table of numbers, using $1,2,3,\ldots,n^2$ such that every row, every column, and both diagonals have the same sum (called the magic number of this square).
A story goes that in medieval times odd order magic squares where the center square is a 1 is special, perhaps even **godly**. So let us call those magic squares whose center square is 1 a **godly magic square**.....(read more)# General position in chess: No-three-in-line problem.
https://radiofreemath.org/read/posts/0+inbox/General+position+in+chess+no+three+in+line+problem.md
2023-05-11 11:51Try this. On an $8\times 8$ chess board, can you put $16$ pawns such that no three of them are in a line? By line we mean ANY line, **not just** horizontal, vertical, or $45^\circ$ lines.
We often say a set of points in the plane is in **general position** if no three points are in a line. So here we are looking for general positions of pawns in an $n\times n$ chess board. Given $n$, what is the **largest number** of pieces you can put in general position on an $n\times n$ chess board?....(read more)# Guessing a natural polynomial.
https://radiofreemath.org/read/posts/0+inbox/Guessing+a+natural+polynomial.md
2023-05-11 11:51Suppose you are given a mysterious polynomial $p(x) \in \mathbb N_0 [x]$, where we do not know its degree nor any coefficients, except we do know its coeffients are non-negative integers.
You may ask to evaluate $p$ at any input $x$ of your choosing, and you will be given the value of $p(x)$. In how many tries can you completely determine this polynomial $p$? Do it in the least possible tries possible.....(read more)# Frodo's number.
https://radiofreemath.org/read/posts/0+inbox/Frodos+number.md
2023-05-11 11:51Frodo thinks of a positive integer $N$ that satisfies the following properties
(1) The sum of the digits in $N$ equals to the product of the digits in $N$.
(2) Exactly half of the digits in $N$ are $1$'s.....(read more)# $G/Z(G)$ cyclic implies $G$ is abelian.
https://radiofreemath.org/read/posts/0+inbox/G+mod+Z(G)+cyclic+implies+G+abelian.md
2023-05-11 11:51This is a classic exercise, but has a cute application in the [[0 inbox/0.625 theorem|5/8 theorem]]. And since when $G$ abelian, the center $Z(G)=G$, this implies either $G/Z(G)$ is trivial (when $G$ abelian), or $G/Z(G)$ non-cyclic (when $G$ non-abelian).
Recall the center $Z(G)=\{g\in G: xg=gx \text{ for all }x\in G\}$, namely the set of all elements in $G$ that would commute with any element in $G$. This is a normal subgroup, so the quotient $G/Z(G)$ is well-defined. ....(read more)# Four intersecting quarter circles.
https://radiofreemath.org/read/posts/0+inbox/Four+intersecting+quarter+circles.md
2023-05-11 11:51A friend during highschool, after learning about **integration** in calculus, told me that the following shaded area can only be found using calculus, the intersection of four quarter circles in a unit square:
![[---images/Four intersecting quarter circles 2023-05-04 14.10.30.excalidraw.svg]]
At the time I had not taken calculus yet. So naturally, I decided to do so with *basic geometry* only, which was **possible**. ....(read more)# Center of quarternion group.
https://radiofreemath.org/read/posts/0+inbox/Center+of+quaternion+group.md
2023-05-11 11:51Consider the order 8 quarternion group $Q=\{\pm 1, \pm i \pm j, \pm k\}$ with the relations $i^2 = j^2 = k^2 = ijk = -1$, and $(-1)^2 = 1$. We remark its center is $\{\pm 1\}$, and the quotient $Q/Z(Q) \approx V_4$, the Klein-4 group.
This isn't hard to see, as none of the $\pm i,\pm j,\pm k$ elements commute with each other, while $\pm 1$ does. So $Z(Q) = \{\pm1\}$. So the cosets are just $aZ = \{\pm a\}$, for $a=1,i,j,k$. The nontrivial elements are each order $2$, so this is precisely the Klein-4 group. ....(read more)# Center of dihedral groups.
https://radiofreemath.org/read/posts/0+inbox/Center+of+dihedral+groups.md
2023-05-11 11:51Define the dihedral group $Dih_n$ to be the symmetry group of a regular $n$-gon (and of $2n$ elements). We will show that they have center $Z(Dih_n)$ isomorphic to either $\mathbb Z_2$ or trivial, and compute the quotient $Dih_n/Z(Dih_n)$. Note the center $Z(G)$ of a group $G$ is the set of elements that commutes with everything in $G$.
In particular, if $n\ge4$ is even, then $Z(Dih_n) = \{1, r^{n/2}\}$ and $Dih_n / Z(Dih_n) \approx Dih_{n/2}$. ....(read more)# The 5/8 theorem, and the most abelian non-abelian groups.
https://radiofreemath.org/read/posts/0+inbox/0.625+theorem.md
2023-05-11 11:51Imagine you roll two six-sided dice, each face labeled with one of the elements of $S_3\approx Dih_3 = \{1,r,r^2,f,fr,fr^2\}$. What is the probability that the two elements $x,y$ we get commute, namely $xy=yx$?
If we draw out its multiplication table, we can highlight the outcomes where they commute, and we see that this probability is $1/2$ for two $S_3$ dice. Call this its **commuting probability** $p_{S_3}$. Can we do better? Of course if the group is abelian, then the commuting probability is 1. So we are interested in the commuting probability $p_G$ of a non-abelian group $G$. ....(read more)