Five Ways to Cheat On a Math Test

OUR MATH TEACHER
by: Putra Prima Raka
Nama : Ms. Shani Rahmawati
Tempat tanggal lahir
: Bandung, 26 Oktober 1989
Umur: 23
Alamat Rumah
: Komp. Margahayu Kencana blok A 12 no. 3, Bandung
Tempat mengajar
: SMP – SMA Kharisma Bangsa
Aktif mengajar
: Sejak 9 November 2011
University
: Universitas Pendidikan Indonesia (UPI)
M
s. Shani adalah salah satu guru matematika di SMP dan
SMA Kharisma Bangsa. Beliau mengajar beberapa kelas
yang salah satunya adalah kelas 10A. Ms. Shani merupakan
orang yang baik dan easy going kepada siswanya. Metode mengajar Ms.
Shani, dalam menjelaskan suatu materi adalah dengan memberikan
pemahaman terlebih dahulu kepada siswa, kemudian perlahan-lahan
terbentuklah sebuah rumus. Ibarat sebuah pohon, dibayagkan sebuah
akarnya, lalu ditampakkan pohonnya.
Ms. Shani sendiri pernah menuangkan prestasi di Ranking 1,
acara kuis pelajaran yang diadakan di Trans TV. Ms. Shani dapat
menyelesaikan semua pertanyaan yang diberikan. Pertanyaan tentang
Pearl Harbour, paham animisme, hingga penemu antibiotic penicillin
pun mampu dijawab dengan percaya diri. Namun di saat pertanyaan
bergulir tentang pengetahuan astronomi yaitu supernova ledakan di galaksi yang dapat menghasilkan energi yang
lebih besar dari energi matahari mengandaskan harapan Ms. Shani Rahmawati untuk masuk ke babak berikutnya.
Walaupun begitu, Ms. Shani merupakan orang terakhir yang tersisa dan menjadi pemenang yang mendapatkan
plakat khusus Ranking 1. Bravo Ms. Shani !
M2: Math Magazine | 01
ARCIMHDHS
By: Inggita Pramesthi Ananda
Nama
: Archimedes of Syracuse
Yunani: Aρχιμηδης
Lahir
: 287 BC Syracuse, Sicily
Wafat
: 212 BC (around 75)
Terkenal untuk:
- Archimedes' principle
- Archimedes' screw
- hidrostatik
- levers
- infinitesimals
A
rchimedes dulu seorang ahli matematika,
fisika, insinyur dan astronomi. Dia adalah
salah satu ahli matematika yang paling terkenal
dan hebat. Dia juga menemukan pi yang akurat ketika
ia menggunakan metode-metode untuk mencari luas
parabola. Selain itu dia juga menemukan cara untuk
menulis angka-angka besar. Dia berhasil membuktikan
bahwa volume dan luas permukaan bola (sphere) adalah
dua per tiganya dari silinder. Dia meninggal ketika
ia dbunuh tentara Roma walau perintahnya adalah
Archimedes tidak disakiti.
bertambah sama dengan badan dia yang masuk ke
bak. Saat itu juga ia lari ke istana raja telanjang sambil
berteriak “EURIKA”. Ia menemukan density.
Archimedes’s screw: dulu digunakan sebagai alat
untuk memindahkan air dari dataran rendah ke tempatArchimedes’s Principle: adalah hukum tentang tempat untuk irigasi. Bentuknya hampir seperti bor.
prinsip pengapungan diatas benda cair yang berbinyi Desainnya adalah semacam sekrup yang dimasukan
“Suatu benda yang dicelupkan sebagian atau seluruhya ke dalam pipa yang diputar secara manual yang dapat
kedalam zat cair akan mengalami gaya ke atas yang mengangkat air.
besarnya sama dengan berat zat cair yang dipindahkan
Hidrostatik: adalah alat-alat yang ditemukan
oleh benda tersebut” (rumus: FA = ρa . Va . g)
Archimedes yang mendalami fluida tak bergerak. Ini
Archimedes menemukan teori ini ketika ia
diperintahkan raja untuk mengetahui apakah mahkotanya
100% emas atau ada peraknya tetapi Archimedes
tidak boleh merusak atau membongkar mahkotanya.
Setelah berfikir beberapa lama ia mandi. Archimedes
sadar bahwa ketika ia masuk ke bak mandinya air yang
02 | M2: Math Magazine
adalah tekanan yang diakibatkan oleh gaya yang ada
pada zat cair terhadap suatu luas bidang tekan pada
kedalaman tertentu. Besarnya tekanan ini bergantung
kepada ketinggian zat cair, massa jenis dan percepatan
gravitasi. Tekanan Hidrostatika hanya berlaku pada zat
cair yang tidak bergerak.
and scientist, but he is best known for the Pythagorean
theorem which bears his name. However, because
legend and obfuscation cloud his work even more
than that of the other pre-Socratic philosophers, one
can give only a tentative account of his teachings, and
some have questioned whether he contributed much
to mathematics and natural philosophy. Many of the
accomplishments credited to Pythagoras may actually
have been accomplishments of his colleagues and
successors. Whether or not his disciples believed that
everything was related to mathematics and that numbers
were the ultimate reality is unknown. It was said that he
was the first man to call himself a philosopher, or lover
of wisdom,[3] and Pythagorean ideas exercised a marked
influence on Plato, and through him, all of Western
philosophy.
Pythagorean theorem
Πυθαγορας
P
By: Happy Hariani
ythagoras of Samos (Ancient Greek:
Πυθαγορας ο Σαμιος [Πυθαγορης in Ionian
Greek] Pythagóras ho Sámios "Pythagoras the
Samian", or simply Πυθαγορας; b. about 570 – d.
about 495 BC) was an Ionian Greek philosopher,
mathematician, and founder of the religious movement
called Pythagoreanism. Most of the information about
Pythagoras was written down centuries after he lived,
so very little reliable information is known about him.
He was born on the island of Samos, and might have
travelled widely in his youth, visiting Egypt and other
places seeking knowledge. Around 530 BC, he moved to
Croton, a Greek colony in southern Italy, and there set
up a religious sect. His followers pursued the religious
rites and practices developed by Pythagoras, and studied
his philosophical theories. The society took an active role
in the politics of Croton, but this eventually led to their
downfall. The Pythagorean meeting-places were burned,
and Pythagoras was forced to flee the city. He is said to
have died inMetapontum.
Pythagoras made influential contributions to
philosophy and religious teaching in the late 6th century
BC. He is often revered as a greatmathematician, mystic
Since the fourth century AD, Pythagoras has
commonly been given credit for discovering the
Pythagorean theorem, a theorem in geometry that states
that in a right-angled triangle the area of the square on
the hypotenuse (the side opposite the right angle) is equal
to the sum of the areas of the squares of the other two
sides—that is, .
While the theorem that now bears his name was
known and previously utilized by the Babylonians
and Indians, he, or his students, are often said to
have constructed the first proof. It must, however, be
stressed that the way in which the Babylonians handled
Pythagorean numbers implies that they knew that the
principle was generally applicable, and knew some
kind of proof, which has not yet been found in the (still
largely unpublished) cuneiform sources. Because of
the secretive nature of his school and the custom of its
students to attribute everything to their teacher, there is
no evidence that Pythagoras himself worked on or proved
this theorem. For that matter, there is no evidence that
he worked on any mathematical or meta-mathematical
problems. Some attribute it as a carefully constructed
myth by followers of Plato over two centuries after
the death of Pythagoras, mainly to bolster the case for
Platonic meta-physics, which resonate well with the
ideas they attributed to Pythagoras. This attribution has
stuck down the centuries up to modern times.The earliest
known mention of Pythagoras's name in connection with
the theorem occurred five centuries after his death, in the
writings of Cicero and Plutarch.
M2: Math Magazine | 03
By: Indah Ftriani
A Games of Shadow
Sherlock Holmes
Mathematics in
A
n evil mastermind is set on bringing about
global war. Only one man can stop him:
Sherlock Holmes, with the help of his partner
in crime-solving Dr Watson. But in the latest Holmes
flick, Sherlock Holmes: A game of shadows, they don’t
just need their trusty revolvers and Holmes’s trademark
prescient fight scenes, they also need to grasp some
mathematics.
with equations. Not only did they have to be real, they
had to be historically accurate, based on a 19th-century
understanding of the field.
“When we did the equations on the blackboard, [the
film-makers] got excited,” says Goriely. “Although they
were quite secretive about the story, they told us that
Moriarty was a mathematics professor and that they
wanted us to help them add more meat to the script,
which was a little dumb and mostly incorrect.”
The villain is Holmes’s nemesis, James Moriarty, a
professor of mathematics and all-around evil genius. In
Goriely and Moulton ended up going beyond scriptthe book The Final Problem, he is described by Holmes tweaking to develop a secret code from scratch that
himself as “a genius, a philosopher, an abstract thinker. Moriarty uses in the film to send messages around a
He has a brain of the first order.”
Europe on the brink of the war he is conniving.
But behind the wit of the character in the film lies the
But how does one get into the mindset of a fictional
mathematical know-how of a team at the University of evil genius from the 19th century? Unfortunately,
Oxford. Alain Goriely and Derek Moulton at Oxford’s Arthur Conan Doyle’s books were of limited help,
Mathematical Institute have been hard at
offering sparse details on Moriarty’s interests, Goriely
work behind the scenes helping to formulate a
believable mathematical villain.
Initially, the filmmakers approached the
mathematicians to ask them to fill Moriarty’s blackboard
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says. “We do know that the character wrote two books one on binomial theorem and one titled The Dynamics
of an Asteroid.”
To create a convincing code, the team started from
the binomial theorem. “Binomial theorem is linked to build a missile and throw it out of the atmosphere, it
Pascal’s triangle, so we devised a secret code based on could re-enter with an asteroid-like impact. It would be
brought back by gravitational forces,” he says.
that,” says Gorielyj.
The lecture discusses the n-body problem - a
mathematical problem that considers how moving
celestial bodies interact with each other as a result of
their gravitational energy. Moriarty would have likely
had a particular interest in the theory, given its potential
implications for weaponry, says Goriely. “If you could
build a missile and throw it out of the atmosphere, it
could re-enter with an asteroid-like impact. It would be
The pair also wrote an entire lecture for Moriarty brought back by gravitational forces,” he says.
While a disguised Holmes might have been party
based on his interests in celestial dynamics. “I used
elements of maths from celestial mechanics at the end of to the entirety of the lecture, sadly only the smallest of
the 19th century,” says Goriely. “It was a very hot topic snippets made the final cut for the audience’s edification.
And while Holmes’s fleeting glance of Moriarty’s
at the time.
” The lecture discusses the n-body problem - a blackboard proved key to his later success in foiling
mathematical problem that considers how moving the professor’s evil plans, even the most beady-eyed
celestial bodies interact with each other as a result of mathematician watching the film will find such a feat
their gravitational energy. Moriarty would have likely tricky. But perhaps therein lies the attraction of Sherlock
had a particular interest in the theory, given its potential Holmes and his amazing powers of deduction.
The code is hidden in Moriarty’s red pocketbook,
which is filled with numbers. The numbers signal to the
reader first which Fibonacci p-code - a way to take digits
from Pascal’s triangle - to use. This supplies another list
of numbers, which are used to indicate which page, line
and words from a book to look up. Goriely reckons his
code is spot on for the character. “Moriarty was obsessed
with Pascal’s triangle and Fibonacci’s codes,” he says.
implications for weaponry, says Goriely. “If you could
M2: Math Magazine | 05
Copernicus
Nicolaus
by: Kausalya Frida Devara
"To know that we know
what we know, and to know
that we do not know what
we do not know, that is true
knowledge."
N
icolaus Copernicus was born on February
19, 1473 in Torun, Poland. Circa 1508,
Copernicus developed his own celestial model
of a heliocentric planetary system. Around 1514, he
shared his findings in the Commentariolus. His second
book on the topic,De revolutionibus orbium coelestium,
was banned by the Roman Catholic Church not long
after his May 24, 1543 death in Frauenburg, Poland.
BEST KNOWN FOR
Astronomer Nicolaus Copernicus identified
the concept of a heliocentric solar system,
in which the sun, rather than the earth, is the
center of the solar system.
Famed astronomer Nicolaus Copernicus (Mikolaj
Kopernik, in German) came into the world on February
19, 1473. The fourth and youngest child born to Nicolaus
Copernicus Sr. and Barbara Watzenrode, an affluent
copper merchant family in Torun, Poland
In 1491, Copernicus entered the University of
Cracow, where he studied painting and mathematics
Upon graduating from Cracow in 1494, Copernicus
returned to Torun, where he took a canon's position—
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arranged by his uncle—at Frombork's cathedral.
In 1496, Copernicus took leave and traveled to
Italy, where he enrolled in a religious law program as
the University of Bologna. There, he met astronomer
Domenico Maria Novara—a fateful encounter, they
decided to become roommates.
In 1500, after completing his law studies in Bologna,
Copernicus went on to study practical medicine at the
University of Padua.. In 1503, Copernicus attended the
University of Ferrara, After passing the test on his first
attempt, he hurried back home to Poland .
Copernicus remained at the Lidzbark-Warminski
residence for the next seven years, working and tending
to his elderly, ailing uncle, and exploring astronomy
whenever he could find the time.
In 1510, Copernicus moved to a residence in the
Frombork Cathedral Chapter in hopes of clearing
additional time to study astronomy. He would live there
as a canon for the duration of his life.
Vieta’s Theorem
T
by: Raisya Bilqis Seruni
hese formulas were discovered on the 16th century by French
mathematician François Viète or he is known as Franciscus
Vieta. He was a French lawyer and mathematician. The
formula named after the inventor itself, that is why it is called as
Vieta’s Theorem.
Do you even know what Vieta’s Theorem is? Here is some
explanation about it!
Vieta's Formulas can be used to relate the sum and product of the
roots of a polynomial to its coefficients. Vieta's formulas give a simple
relation between the roots of a polynomial and its coefficients. In the
case of the quadratic polynomial, they take the following form:
to find the sum of the roots
and
to find the product of the roots
M2: Math Magazine | 07
INTEGRAL
I
by: Ilham Zaky Wilson
ntegral is derivative’s opponent. Integral is a very graph of f, the x-axis, and the vertical lines x = a and x = b.
important concept in mathematics.
Figure 1: A definite
It enables us to find the origin of a function, calculate
the volume and the area of curve. Together with its
inverse, differentiation is one of the two main operations
in calculus.
∫f(X)dx=F(x)+C
integral of a function
can be represented as
the signed area of the
region bounded by its
graph.
∫ = Integral sign which declares anti-differential
operation.
f(X) = Integral function, a function from which an
anti-differential can be found.
dx = Differential
C = Constant
Integral has numerous applications in science and
mathematics field. It is used as the main tool (together
with differential) in calculus. The founders of the
calculus thought of the integral as an infinite sum of
rectangles of infinitesimal width.
Given a function f of a real variable x and an interval
Bernhard Riemann gave the rigorous description
[a, b] of the real line, the definite integral is defined
of integral. It is based on a limiting procedure which
informally to be the area of the region
approximates the area of a curvilinear region by breaking
in the Cartesian-plane bounded by the
the region into thin vertical slabs. Beginning in the 19th
08 | M2: Math Magazine
century, more sophisticated notions of integrals began
to appear, where the type of the function as well as the
domain over which the integration is performed has been
generalised.
So, we’ll get: this:
du/dx=u' -> this: du/u'=dx.
Now that u' is the derivative of (2x3-5).
We’ll get:
A line integral is defined for functions of two or three
du/6x2
variables, and the interval of integration [a, b] is replaced
Then, substitute our new differential sign into our
by a certain curve connecting two points on the plane or
in the space. In a surface integral, the curve is replaced current integral function.
by a piece of a surface in the three-dimensional space.
Integral problems:
Integrals of differential forms play a fundamental role in
modern differential geometry.
These generalizations of integrals first arose from
the needs of physics, and they play an important role in
the formulation of many physical laws, notably those of
We can cross and change several notations, so that
electrodynamics. There are many modern concepts of we’ll get a simpler function.
integration, among these; the most common is based on
the abstract mathematical theory known as Lebesgue
integration, developed by Henri Lebesgue.
Integral’s main formula is described by this simple
general equation:
Pull out 3 from our integral function:
Sample problem:
Finally, perform our lovely anti-differential operation!
Solution:
Let
Here’s our result:
u=(2x3-5)
Then substitute u into the equation.
In here, we get 2 different variable:
u & x.
Now, let’s focus on x. We’ve to replace dx with du.
It’ll be like this:
du/dx=u'
Then, swap the position of u^'& dx.
Since
u=(2x3-5)
M2: Math Magazine | 09
Why is It So Special?
Euler's identity is considered by many to be
remarkable for its mathematical beauty. These three basic
arithmetic operations occur exactly once each: addition,
multiplication, and exponentiation. The identity also
links five fundamental mathematical constants. We are
going to discuss them one by one.
1. Imaginary Number
The number i, the imaginary unit of the complex
numbers, a field of numbers that contains the roots of all
polynomials (that are not constants), and whose study
leads to deeper insights into many areas of algebra and
calculus, such as integration in calculus.
The unit imaginary number, i,
equals the square root of minus 1
Imaginary Numbers are not "imaginary", they really
exist, and you may need to use them one day.
The Unit Imaginary Number, i, has an interesting
property. It "cycles" through 4 different values each time
you multiply:
Discover The Most Wonderful Equation:
“Euler’s Identity”
by: Radryan Andrayukti
What is It?
L
eonhard Euler is a Swiss mathematician , doing
a really long process, has found and proved a
fantastic equation called “Euler’s Identity” or
“Euler’s Equation”. Many experts, mathematicians,
physicians, even universities were astonished by this
equation. From Paul Nahin (a professor emeritus at the
University of New Hampshire), the mathematician Carl
Friedrich Gauss, Benjamin Peirce ( a noted American
19th-century philosopher, mathematician, and professor
at Harvard University), until Stanford University
mathematics professor Keith Devlin stated many words
like "the gold standard for mathematical beauty" and "it
is absolutely paradoxical; we cannot understand it, and
we don't know what it means, but we have proved it, and
therefore we know it must be the truth.".
i × i = -1,
then -1 × i = -i,
then -i × i = 1,
then 1 × i = i
(back to i again!)
2. Euler Number
The number e is a famous irrational
number, and is one of the most
important numbers in mathematics.
The first few digits are:
2.718281828459045235360287
4713527 (and more ...)
It is often called Euler's number after Leonhard
Eulere is the base of the Natural Logarithms (invented by
John Napier). On the other hand Common Logarithms
have 10 as their base.
Often the number e appears in unexpected places
and can really be used in our life.
For example, e is used in Continuous Compounding
(for loans and investments):
Here’s the form of Euler’s Identity:
Formula for Continuous Compounding
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1+0=1
234 + 0 = 234
3. Phi
Draw a circle with a radius of 1.
The distance half way around the edge of the circle
will be 3.14159265... a number
known as Pi.
Or you could draw a circle
with a diameter of 1. Then the
circumference (the distance all the
way around the edge of the circle)
will be Pi.
Pi (the symbol is the Greek letter π)
is:
The ratio of the Circumference to
the Diameter of a Circle.
In other words, if you measure the circumference,
and then divide by the diameter of the circle you get the
number π
The history of the number 0 might start when AlKhwarizmi wrote a treatise on Hindu-Arabic numerals.
The Arabic text is lost but a Latin translation, Algoritmi
de numero Indorum in English Al-Khwarizmi on the
Hindu Art of Reckoning gave rise to the word algorithm
deriving from his name in the title. Unfortunately the
Latin translation (translated into English in [19]) is
known to be much changed from al-Khwarizmi's original
text (of which even the title is unknown). The work
describes the Hindu place-value system of numerals
based on 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0. The first use of
zero as a place holder in positional base notation was
probably due to al-Khwarizmi in this work. Methods for
arithmetical calculation are given, and a method to find
square roots is known to have been in the Arabic original
although it is missing from the Latin version.
5. One
Everybody knows the number one is the multiplicative
identity. Every number multiplied by 1 becomes the
number itself.
x1=
22
4545
369369
56235623
9982599825
SUDOKU
by: Isabella Regia
It is approximately equal to:
3.14159265358979323846…
The digits go on and on with no pattern. In fact, π has
been calculated to over two quadrillion decimal places
and still there is no pattern.
4. Zero
The number 0 is the additive identity. It means when
we add any number with zero it becomes the number
itself.
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1. Read ahead
Read over tomorrow’s math lesson today. Get a general idea about the new
formulas in advance, before your teacher covers them in class.
As you read ahead, you will recognize some of it, and other parts will be brand new.
That’s OK – when your teacher is explaining them you already have a “hook” to
hang this new knowledge on and it will make more sense — and it will be easier to
memorize the formulas later.
3. Keep a list of symbols
Most math formulas involve some Greek letters, or perhaps some strange symbols
like ^ or perhaps a letter with a bar over the top.
When we learn a foreign language, it’s good to keep a list of the new vocabulary as
we come across it. As it gets more complicated, we can go back to the list to remind
us of the words we learned recently but are hazy about. Learning mathematics
symbols should be like this, too.
Keep a list of symbols and paste them up somewhere in your room, so that you can
update it easily and can refer to it when needed. Write out the symbol in words, for
example: ∑ is “sum”; ∫ is the “integration” symbol and Φ is “capital phi”, the Greek
letter.
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by: Nadira (Dea)
I
n the history of mathematics, mathematics in
medieval Islam, often called Islamic mathematics or
Arabic mathematics, covers the body of mathematics
preserved and advanced under the Islamic civilization
between circa 622 and c.1600. Islamic science and
mathematics flourished under the Islamic caliphate
established across the Middle East, extending from the
Iberian Peninsula in the west to the Indus in the east and
to theAlmoravid Dynasty and Mali Empire in the south.
In his A History of Mathematics, Victor Katz says
that:
A complete history of mathematics of medieval
Islam cannot yet be written, since so many of
these Arabic manuscripts lie unstudied... Still, the
general outline... is known. In particular, Islamic
mathematicians fully developed the decimal placevalue number system to include decimal fractions,
systematised the study of algebra and began to
consider the relationship between algebra and
geometry, studied and made advances on the
major Greek geometrical treatises of Euclid,
Archimedes, andApollonius, and made significant
improvements in plane and spherical geometry.
An important role was played by the
translation and study of Greek mathematics,
which was the principal route of transmission of
these texts to Western Europe. Smith notes that:
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The world owes a great debt to Arab scholars for
preserving and transmitting to posterity the classics
of Greek mathematics... their work was chiefly that of
transmission, although they developed considerable
ingenuity in algebra and showed some genius in their
work in trigonometry.
Adolph P. Yushkevich states regarding the role of
Islamic mathematics:
The Islamic mathematicians exercised a prolific
influence on the development of science in Europe,
enriched as much by their own discoveries as those they
had inherited by the Greeks, the Indians, the Syrians, the
Babylonians,etc.
by: Dellamartha Anjani
T
he Rubiks Cube is a cube consisting of 6 sides
with 9 individual pieces on each. The main
objective when using one is to recreate its
original position, a solid color for each side, without
removing any piece from the cube. Though it is colorful
and looks like a children's toy, there have been many
championships for its completion. It amused five yearolds yet inspired mathematicians.
The number of possible positions of Rubik's cube is:
The cube has even been used in college math classes
dealing with group theory, a branch of algebra having to
do with geometric symmetry developed in the nineteenth
century. Group theory shows that a 60 degree rotation
of a six-pointed snow flake makes the flakes appearance
unchanged. Each group theory is symmetrical, and the
cube represents this is after rotation.
Its unique design was made by an engineer named
Erno Rubik, a socialist bureaucrat who lived in Budapest,
Hungary. He built the simple toy in his mother's
apartment and did not know of the 500 million people
who were going to become overly perplexed over it. His
first idea of the cube came in the spring of 1974.
The cube can be solved in two ways. One can use
sequences to solve piece by piece, or you can attempt
to solve it backwards. This means that after the cube is
completed and mixed, you can figure what turns were
made to mix it and undo them. Mathematicians have
tried to find the shortest method of unscrambling, which
What inspired Erno was the popular puzzle before became known as God's algorithm.
his called the 15 Puzzle. Invented in the late 1870's,
God's algorithm relies on a tree structure of all possible
this puzzle consisted of 15 consecutively numbered, flat scrambled positions, where a node is a position found
squares that can be slid around inside a square frame. Sam by making a move to scramble the cube from a previous
Loyd created this two dimensional version of the Rubiks node. The root of the tree is the single initial position
Cube. The puzzle was originally called the Magic Cube where the cube is solved. The algorithm searches for the
-- or Buvuos Kocka in Hungarian. It was later renamed in matching scrambled position from the root of the tree
honor of its creator to the Rubiks Cube. Many different and a solution is found by traversing the actions leading
cube variations have been made, but the simplest one is to the path found. Although God's algorithm is fast, it is
called the standard 3x3x3. It contains 26 little blocks of more of a computing approach rather than mathematical
plastic.
approach.
The Rubiks Cube has been a successful product for
many years. Though created without great intentions,
people have spent millions of dollars on it. Math classes
to this day study the complexity of the Cube. Erno, the
creator of the cube, became an overly rich man from his
ingenious creation.
The cube can rotate around its center in any way
possible. No pieces are restricted to any singular
movement. The cube is not easily solved because it does
not have a definite scrambled point. This means that
there is only one completed situation, where all the sides
have one color each.
M2: Math Magazine | 15
By: Al Farisi Firdaus
Eratosthenes
The Sieve Of
I
n mathematics, the sieve of Eratosthenes (Greek:
κοσκινον Eρατοσθeνους), one of a number of
prime number sieves, is a simple, ancient algorithm
for finding all prime numbers up to any given limit. It does
so by iteratively marking as composite (i.e. not prime) the
multiples of each prime, starting with the multiples of 2.
Otherwise, let p now equal this number (which is the
next prime), and repeat from step 3.
When the algorithm terminates, all the numbers in
the list that are not marked are prime.
The main idea here is that every value for p is prime,
because we have already marked all the multiples of the
numbers less than p.
As a refinement, it is sufficient to mark the numbers
in step 3 starting from p2, as all the smaller multiples of p
will have already been marked at that point. This means
that the algorithm is allowed to terminate in step 4 when
p2 is greater than n.
Another refinement is to initially list odd numbers
only, (3, 5, ..., n), and count up using an increment of 2p in
step 3, thus marking only odd multiples of p greater than
p itself. This actually appears in the original algorithm.
[1] This can be generalized with wheel factorization,
forming the initial list only from numbers coprime with
the first few primes and not just from odds, i.e. numbers
coprime with 2.
The multiples of a given prime are generated starting
Example
from that prime, as a sequence of numbers with the same
To find all the prime numbers less than or equal to
difference, equal to that prime, between consecutive
numbers.[1] This is the sieve's key distinction from using 30, proceed as follows.
trial division to sequentially test each candidate number
First generate a list of integers from 2 to 30:
for divisibility by each prime.
First number in the list is 2; cross out every 2nd
The sieve of Eratosthenes is one of the most efficient number in the list after it (by counting up in increments
ways to find all of the smaller primes (below 10 million of 2), i.e. all the multiples of 2:
or so).It is named after Eratosthenes of Cyrene, a Greek
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
23
24
25 26 27 28 29 30
mathematician; although none of his works have survived,
Next number in the list after 2 is 3; cross out every 3rd
the sieve was described and attributed to Eratosthenes in
number in the list after it (by counting up in increments
the Introduction to Arithmetic by Nicomachus.
of 3), i.e. all the multiples of 3:
Algorithm Description
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
A prime number is a natural number which has 23 24 25 26 27 28 29 30
exactly two distinct natural number divisors: 1 and itself.
Next number not yet crossed out in the list after 3
To find all the prime numbers less than or equal to a is 5; cross out every 5th number in the list after it (by
given integer n by Eratosthenes' method:
counting up in increments of 5), i.e. all the multiples of 5:
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
Create a list of consecutive integers from 2 to n: (2,
23 24 25 26 27 28 29 30
3, 4, ..., n).
Next number not yet crossed out in the list after 5 is
Initially, let p equal 2, the first prime number.
7; the next step would be to cross out every 7th number in
Starting from p, count up in increments of p and the list after it, but they are all already crossed out at this
mark each of these numbers greater than p itself in the point, as these numbers (14, 21, 28) are also multiples
list. These will be multiples of p: 2p, 3p, 4p, etc.; note of smaller primes because 7*7 is greater than 30. The
that some of them may have already been marked.
numbers left not crossed out in the list at this point are all
Find the first number greater than p in the list that is the prime numbers below 30:
not marked. If there was no such number, stop.
2 3 5 7
11 13
17 19
23
29
2
16 | M : Math Magazine
Ulam Spiral
By: Syafira Alyani
he Ulam spiral, or prime spiral (in other This is a 200×200 Ulam spiral, where primes are black.
languages also called Ulam cloth) is a simple Black diagonal lines are clearly visible.
method of graphing the prime numbers that
reveals a pattern which has never been fully explained. It
was discovered by themathematician Stanislaw Marcin
Ulam in 1963, while doodling on scratch paper at a
scientific meeting. Ulam, bored that day, wrote down a
regular grid of numbers, starting with 1 at the center, and
spiraling out like this.
T
It appears that there are diagonal lines no matter
how many numbers are plotted. This seems to remain
true, even if the starting number at the center is much
larger than 1. This implies that there are many integer
constants b andc such that the function:
He then circled all of the prime numbers and he got
the following picture:
f(n) = 4 n2 + b n + c
generates an unexpectedly-large number of primes as
n counts up {1, 2, 3, ...}. This was so significant, that the
Ulam spiral appeared on the cover of Scientific American
in March 1964.
At sufficient distance from the centre, horizontal and
vertical lines are also clearly visible.
To his surprise, the circled numbers tended to line up
along diagonal lines. The following image illustrates this.
M2: Math Magazine | 17
~~Anak IMO??? Atau anak EMO? :P~~
By: Wani Riselia Sirait
I
nternational Mathematics Olympiad or
IMO is the World Championship
Mathematics
Competition
for
High
School students. It is also the oldest of the
International Science Olympiad.
The first IMO was held in 1959 in Romanya, and
was participated by 7 countries. It is held annually
(except in 1980) in a different country. So, this
year is the 54th IMO and it will be held in Santa
Marta, Colombia from the 18th –28th of July of
2013.
About 100 countries send team up to six students. Actually, every
contestant can join IMO before they enter university, like Terence Tao
(Australia) who
won bronze,
silver, and gold
medal
respectively.
He won a gold
medal when he
just turned
thirteen in IMO
18 | M2: Math Magazine
Terence Tao
Reid Barton 1988 and becoming
M2: Math Magazine | 19
20 | M2: Math Magazine
Mental Abuse To Humans
Five Ways to Cheat On a Math Test
By: Azkia Rahmah
Math can be hard. If you get overwhelmed and decide to turn to cheating, this may work for you.
1. Calculator Method
2. Mobile Phone Method

text someone

look up an example solution in the Internet
3.
Tapping Method

Find a friend near you that knows mathematics well.

Make a tapping code
4. Pen Method

Slide a rolled math note into a pen that you can unscrew for removing and
replacing ink cartridges.
5. Other Methods

Try pretending to scratch your head while looking next to you. By having
your hand and arm to your head, the teacher won't see your eyes shifting to
someone's paper.

Write formulas near you. This could include clothing, your desk, the cover of
your calculator, your socks, your shoes, or your hand.
Warnings:

God knows and sees everything we do.

Cheating all the time can ruin your future. Do not use these methods entirely as
an excuse never to study and do your work.

Remember that cheating on a test is prohibited by all teachers, and being
caught can cause you to lose your test or a grade reduction.
Tips:


Studying is the best way to do well on a test! Your friend can tell you the answer but
remember—on most math tests you have to explain your work.
If you get stuck solving a problem, think about something completely different and
suddenly the solution will show up.
M2: Math Magazine | 21
22 | M2: Math Magazine
by: Rakha Muhammad Adiyoga
M2: Math Magazine | 23
By: Zacky Rizano, Septadiga Rozamel, Rayhan Fadhila
Papam
K elas 11 A
Menurut gua, pelajaran matematika itu kadang gampang
kadang sulit. Kalo mood gua lagi pengen ngerjain ya
soalnya gampang. Tapi kalau mood gua males ngerjainnya
soalnya biasanya susah.Jadi tergantung mood gua aja.
Verdias
K elas 10 D
Menurut gua, matematika itu susah susah gampang
bukan gampang gampang susah. Lebih banyak susahnya
dibanding gampangnya.
Pak H ikm at
G uru Bahasa Indonesia
Pelajaran matematika itu bagus. Bisa dibilang lebih
bagus dari pelajaran lainnya seperti pelajaran bahasa
Indonesia.Salah satu sebabnya adalah pelajarannya itu
memiliki hasil yang pasti karena sudah ada rumusnya
tidak seperti bahasa Indonesia yang jawabannya itu
rata rata opini.
24 | M2: Math Magazine
M2: Math Magazine | 25
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: Jl. Terbang Layang no 21, Pondok Cabe, Tangsel, 15418
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