(This is sometimes known as the Friendship Theorem, a Theorem on Friends and Strangers, or Ramsey's Theorem in honor of the Cambridge University mathematician Frank Plumton Ramsey who died in 1930 when only 26.)

The problem is naturally modelled on a 2-color graph. To remind, K_n is the standard notation for a complete graph with n vertices. Assume the edges of K₆ are all colored into two colors: red or blue. The problem is equivalent to the statement that for any such coloring there is always a monochromatic triangle (a K₃). In fact, it has been shown by A. W. Goodman (1959) that the number of monochromatic triangles is at least 2!

The applet allows to experiment with the problem and its generalizations. To draw an edge, click on one of the nodes, move the cursor to another node, and click there too. If the AutoColor box is checked, the edges are drawn with alternating colors. Otherwise, you can choose color to sraw an edge with by checking either Red or Blue box.

On K₅ there exist bichromatic colorings without monochromatic K₃ subgraphs. (Think of five people sitting by a round table such that each of them is friends only with his immediate neighbors.) Therefore, 6 is the smallest number k with the property that, for any bichromatic coloring, K_k contains either red K₃ or blue K₃. This is written as

(Note that the Friendship Theorem states that R(3, 3) ≤ 6.) In general, the Ramsey number R(n, m) is the smallest number N such that, for any bichromatic coloring of K_N, there is either red K_n or blue K_m. As an example, it is obvious that, for any m > 1, R(2, m) = m. Obviously, R(m, n) = R(n, m).

It is also known that

The latter result has been established in 1993 and published in 1995. The initial announcement of the result was made, of all places, on the pages of Rochester Democrat and Chronicle. Somebody sent a letter to Ann Landers who published it in her syndicated advisory column. She ended her reply with 'I am "Baffled in Chicago"' [Gardner, pp. 452-453].

The exact value of R(5, 5) is still unknown, except for the bounds

In the 1960s, Gustavus J. Simmons redressed (*) as a game, that became known as the Game of Sim [Gardner, p. 111]. In the game, two players alternate drawing edges of two colors. The loser is the player who first creates a triangle of his or her color. Since (*) has been shown to be correct, Sim can't end in a draw. Since K₆ has 15 edges and 15 is an odd number, the second player has an obvious advantage. However, the game is not quite trivial and several strategies have been published in the 1970s in the Journal of Recreational Mathematics and Mathematics Magazine.

Martin Gardner mentions an observation by F. Harary that (*) affords at least three other types of games. He classified the SIM as the game of avoidance. In the game of achievement the player to complete a monochromatic triangle first wins. This game is trivial. But the other two are not. In the latter games the play continues until all the edges are drawn, after which the players count the number of triangles of their color. The winner may be the player with the most or fewest number of triangles. These last two games are the hardest to analyze.

Helpful in proving (**) is a recursive inequality:

For the proof [Balakrishnan, 1.98], introduce r = R(m-1, n) + R(m, n-1) and consider a company {1, 2, ..., r} of r people. Let M be the set of people known to person 1 and N be set of people not known to person 1. Between them, M and N have r-1 persons. Therefore, either M has at least R(m-1, n) people or N has at least R(m, n-1) people. Assuming the former, M then has at least m-1 people who know each other or n people who do not know each other. This we either have n people who do not know each other or (adding person 1 to M) m people who know each other. Assuming that N has at least R(m, n-1) people we, by symmetry, are led to the same conclusion.

In case R(m-1, n) and R(m, n-1) are both even (1) can be strengthened [Balakrishnan, 1.100]:

To see this, set p = R(m-1, n), q = R(m, n-1) and r = p + q. Consider a group X = {1, ..., r-1} of r-1 people. We wish to show that either X contains m mutual acquaintances or it contains n mutual strangers. Let d_i be the number of people known to person I. Since being an acquaintance is a mutual relationship, the sum Sd_i of all d_i's is even. However, by our assumption, r is even, so that r-1 is odd. It follows that d_i ought to be even for at least one person. For the definiteness sake, suppose d₁ is even and let M and N be the sets of all people known and, respectively, not known to person 1. Denoting the number of elements in a set X as |X|, |M| = d₁, an even number. Since |M| + |N| = r-2, |N| is also even. Further, either M has at least p-1 people or N has at least q people. But p-1 is odd, while |M| is even; thus we can strengthen the above sentence: either M has at least p people or N has at least q people. But if |M| ≤ p (= R(m-1, n)), M then contains either m-1 acquaintances or n strangers. Together with person 1, X has either m acquaintances or n strangers. And we are done in this case. The other is treated similarly.

Thus, for example,

But since both R(3, 3) and R(4, 2) are even, we have a stronger inequality:

To see that R(4, 3) > 8, imagine 8 people sitting at a round table, with each person knowing exactly the two immediate neighbors and the fellow across the table.

By (1),

To see that R(5, 3) > 13, imagine 13 people sitting at a round table such that each person is acquainted only with the fifth person on his left and the fifth person on his right. Under this assumption, there is no 3 mutual acquaintances and no 5 mutual strangers.

R(m, n) with m, n > 2 are known as nontrivial Ramsey numbers. There are also generalizations. Let k_i, i = 1, 2, ..., t, and m be positive integers satisfying k_i ≤ m and t ≤ 2. Let C be the collection of all m-element subsets of an n-element set X. |C| = C(n, m). Let C₁, ..., C_t be an ordered partition of C. Then n is said to possess the generalized (k₁, k₂, ..., k_t; m)-Ramsey property if, for some i, X has a k_i-element subset B such that all m-element subsets of B belong to C_i. The smallest such n is called the generalized Ramsey number, R(k₁, k₂, ..., k_t; m).

R(u, v) = R(u, v; 2). To see that, consider X = {1, 2, ..., n} and the collection C of its 2-element subsets. For a partition C = C₁C₂, think of C₁ and C₂ as containing pairs of mutual acquaintances and mutual strangers. If n = R(u, v), then either X has a group of u acquaintances or a group of v strangers. In the former case, we get a subset B of X, |B| at least u, such that all 2-element subsets of B belong to C₁. In the latter case, a subset B exists with at least v elements, such that all its 2-element subsets belong to C₂. This implies that n = R(u, v) has the generalized R(u, v; 2)-Ramsey property. From the definition it also follows that n is the least such number.

The existence of R(k₁, k₂, ..., k_t; m) is claimed by Ramsey's theorem which is a fundamental part of an extensive Ramsey theory. The theory is concerned with the emergence of certain properties in objects as their scale becomes large. As a rather simple example, the pigeonhole principle is equivalent to the statement:

The Friendship theorem itself has a curious application [Savchev, pp. 146-147, Honsberger, p. 3]. Six random points in space are joined pairwise by 15 line segments. Assume all the segments are of different length. Some segments form triangles. Prove that at least one of the 15 segments serves as the shortest side in one of the triangles it is in, and as the longest side in another. Six points taken by three form 20 triangles. In each, let's color the shortest side in red. It may happen that some of the sides will be painted more than once. Let's color the remaining segments blue. By the Friendship theorem, there is a monochromatic triangle. Since the triangle has a shortest side that had to be colored red, the whole triangle is red. In particular, its longest side is red as well. The fact that it is red means it's the shortest side in some other triangle.

The numbers R(3, 3, ..., 3; m) with m 3s are often denoted R_m. (And sometimes it is m that gets omitted: R(3, 3, ..., 3).) R_m is the minimum number of dots in a complete graph such that if the graph's edges are colored with m colors, there is bound to be at least one monochromatic triangle. R(3, 1) = 3. R(3, 3; 2) = R(3, 3) = 6. R(3, 3, 3; 3) = 17. Although the exact values of R_m are mostly unknown, it can be shown that their values are sharp!

Reference

1. M. Aigner, G. Ziegler, Proofs From The BOOK, Springer, 2000
2. V. K. Balakrishnan, Theory and Problems of Combinatorics, Schaum's Outline Series, McGraw-Hill, 1995
3. M. Gardner, The Colossal Book of Mathematics, W. W. Norton & Co, 2001
4. M. Gardner, Knotted Doughnuts and Other Mathematical Entertainments, W.H.Freeman & Co., 1986.
5. M. Gardner, The Unexpected Hanging and Other Mathematical Diversions, The University of Chicago Press, 1991
6. M. Gardner, The Colossal Book of Short Puzzles and Problems, W. W. Norton & Company, 2006, p. 10
7. A. W. Goodman, On Sets of Acquaintances and Strangers at Any Party, Am Math Monthly, v. 66, no. 9 (Nov., 1959), 778-783.
8. P. Hilton, D. Holton, J. Pederson, Mathematical Vistas, Springer, 2002
9. R. Honsberger, Mathematical Delights, MAA, 2004
10. L. E. Shader, Another Strategy for SIM, Math Magazine, v. 51, No. 1 (Jan., 1978), pp. 60-64
11. S. Savchev, T. Andreescu, Mathematical Miniatures, MAA, 2003

Complex numbers are points in the plane endowed with additional structure. We consider the set R² = {(x, y): x, yR}, i.e., the set of ordered pairs of real numbers. Two such pairs are equal if their corresponding components coincide:

With two operations - addition and multiplication - defined below, the set R² becomes the set C of complex numbers. R² considered as a set of complex numbers C is called Argand diagram or Argand plane or Gauss plane.

Addition

Multiplication

Addition is defined componentwise in a relatively standard way that extends to spaces of higher dimension. Multiplication on the other hand is peculiar to complex numbers. It only has weakened analogues in R⁴ (quaternions) and R⁸ (octonions).

Addition and multiplication of complex numbers inheret most of the properties of addition and multiplication of real numbers:

Several complex numbers play exclusive roles. For example, the number (0, 0) has the properties of 0:

It is therefore natural to identify it with 0. The symbol is exactly the same as used to identify the "real" 0. We shall see shortly that there is a good reason to think of the two zeros - real and complex - as one and the same number.

Another complex number of consequence is (1, 0). This number plays an important role in multiplication that stems from the following property:

Among complex numbers (1, 0) behaves like the real unit 1 among the real numbers. Again, there is a good reason to say that the two are one and the same. It is customary to write (1, 0) = 1.

The third number of importance is (0, 1). It has the remarkable feature of having a negative square. More accurately,

In engineering sciences, the number (0, 1) is sometimes denoted as j. Elsewhere, it is standard to denote it i: i = (0, 1). Multiplication by i has a curious effect:

If you compare two points (x, y) and (y, -x) on the plane (even if for hust a few specific values of a and y) and join the two to the origin, you'll be able to observe that the two segments are perpendicular to each other. Moreover, the latter is obtained from the former by rotation through 90^o in the positive (counterclockwise) direction.

The theory of complex numbers can be developed wholy in algebraic terms, see, for example, Landau. Often, however, both on elementary and advanced level drawing from geometric intuition is extremely useful. Numbers (x, 0) correspond to points on the horizontal x-axis. If it were not for the presence of the y-axis, the points on the horizontal number line would be associated with plain real numbers. The point corresponding to (x, 0) would be considered as a real number x. Thus, it is natural to identify the two representaions of the same point:

Note that previously we already did this for x = 0 and x = 1. The identification (1) is also supported algebraically. For, algebraically, x and (x, 0) are indistinguishable:

as if the second component (0) was not present. With (1) in mind, we can write

which is called the algebraic form of complex number (x, y):

With (1), we easily multiply complex numbers by real:

naturally. Which of course introduces nothing new but a convenience of notations. In the algebraic form, the addition and multiplication are redefined as

Without (1), i² would for ever remain (-1, 0). Taking (1) into account we obtain the famous identity

The possibility of embedding of the set R of reals into the set of complex numbers C, as defined by (1), is probably the single most important property of complex numbers. For, without (1) and (2), the theory of complex numbers would not deliver the closure to the branch of algebra that drove much of its development, viz., the search for the roots of polynomial equations.

The two parts of the complex number z = x + yi have special notations:

Both real and imaginary parts of a complex number are real and any complex number can be written as

Re and Im are mnemonics for Real and Imaginary. Neither is less real or more imaginary than the other. In the complex plane the axes also are referred to as real and imaginary, although both are real enough to the extent that the only way to distinguish between the two is by means of orientation: the rotation from the real to the imaginary axis proceeds counterclockwise. Complex numbers for which the real part is 0, i.e., the numbers in the form yi, for some real y, are said to be purely imaginary.

With every complex number (x + yi) we associate another complex number (x - yi) which is called its conjugate. The conjugate of number z is most often denoted with a bar over it, sometimes with an asterisk to the right of it, occasionally with an apostrophe and even less often with the plain symbol Conj as in

For technical reasons, I prefer using the least common notations, z' or Conj(z):

The importance of the conjugate stems from the following fact:

which, for any complex number z = x + yi, is a non-negative real number. The non-negative square root of this number is called the modulus or absolute value of complex number z:

(m)

.

From the Pythagorean theorem, |z| is the distance from the point represented by z (or the point with complex coordinate z and real coordinates (x, y)) to the origin 0. In general, |z - u| is the distance between points z and u.

The operator Conj is an involution, its square is the identity operator:

and therefore

Also

which easily submits to direct verification. The definition of modulus is consistent with the definition of the absolute value of the real numbers:

In addition, it has several important properties:

The latter is known as the triangle inequality. Geometrically, it's a feature inherited from Euclid. It can be obtained from the Argand diagram, i.e. from the identification of complex numbers with points in a plane with a reference to a triangle inequality valid in the Euclidean plane. It can also be derived algebraically. (m₁) follows from the definition. (m₂) is a consequence of the basic laws

In particular, for a real r > 0, |rz| = r|z|, for any complex z.

Since |z| is the distance to the origin, |z| = 1 is the equation of the unit circle centered at the origin. Points on the unit circle are associated with an angle such that those points and only them have the form (cosa, sina), for some angle a which is not determined uniquely. As a complex number, the points on the unit circle have the form

For any z ≠ 0, |z/|z|| = 1 which means that z/|z| lies on the unit circle and therefore has the above form for some a. If we denote |z| = r, then z can be written as

which is known as the trigonometric form of complex number, its polar representation.

If z ≠ 0 and a₁ and a₂ satisfy (3), then they differ by a factor of 2p:

where k an integer. In other words,

There is a unique a[0, 2p), which denotes a half-open interval from 0 to 2p in which 0 is included but 2p is not. This a is called the argument of z denoted arg(z):

where arg(z)[0, 2p). Zero is not assigned an argument. Thus the argument of a complex number is a real number in a limited interval. The extended argument Arg(z) of a number z is the set of all real numbers congruent to arg(z) modulo 2p:

Arg(z) = {a: a = arg(z) (mod 2p))} = {arg(z) + k·2p: kN}.

So that

z = |z|·(cos(a) + i·sin(a)), for any aArg(z).

Obviously, the number i = (0, 1) plays a special role in the theory of complex numbers. From the properties of sine and cosine, arg(i) = p/2. Furthermore,

Treated as two vectors, (x, y) and (-y, x) are immediately seen to be orthogonal, with the latter obtained from the former by a rotation through p/2 in the positive direction.

(There is a dynamic illustration of the properties of complex numbers and the operations discussed here.)

References

1. T. Andreescu, D. Andrica, Complex Numbers From A to ... Z, Birkhäuser, 2006
2. C. W. Dodge, Euclidean Geometry and Transformations, Dover, 2004 (reprint of 1972 edition)
3. Liang-shin Hahn, Complex Numbers & Geometry, MAA, 1994
4. E. Landau, Foundations of Analisys, Chelsea Publ, 3^rd edition, 1966
5. C. Zwikker, The Advanced Geometry of Plane Curves and Their Applications, Dover, 2005

The applet below illustrates an algorithm for finding the Least Common Multiple (LCM) of a number of integers.

Let there be a finite sequence of positive integers X = (x₁, x₂, ..., x_n), n > 1. The algorithm proceeds in steps as follows: on each step m it examines and updates the sequence X^(m) = (x₁^(m), x₂^(m), ..., x_n^(m)), X⁽¹⁾ = X. The purpose of the examination is to pick the least (perhaps, one of many) element of the sequence X^(m). Assuming x_k0^(m) is the selected element, the sequence X^(m+1) is defined as

In other words, the least element is increased by the corresponding x whereas the rest of the elements pass from X^(m) to X^(m+1) unchanged.

The algorithm stops when all elements in sequence X^(m) are equal. Their common value L is exactly LCM(X).

(In the applet the numbers x₁, x₂, ..., x_n that appear at the top row are modifiable by clicking left or right off their vertical center line. Press button Compute to see the algorithm running. The blue numbers the ones that have been modified at one of the steps.)

Saat menyaksikan tubuh ramping David Gurnani, pemenang The Biggest Loser Asia asal Indonesia, Anda pasti berdecak kagum. Pria berusia 25 tahun ini berhasil menurunkan 70 kg berat badannya (dari 153 kg menjadi 83 kg) dalam waktu lima bulan saja. Meskipun membayangkan sulitnya proses untuk menurunkan berat badan tersebut, dalam hati kecil Anda pasti terbersit keinginan untuk mengikuti program serupa.

Tampaknya, keinginan Anda tersebut bisa segera Anda wujudkan. Fitness First, jaringan klub kebugaran yang menjadi sponsor The Biggest Loser Asia (TBLA), meluncurkan "Lose Big Program". Program penurunan berat badan ini dikembangkan oleh Dave Nuku, Regional Fitness Manager Fitness First yang juga pelatih Tim Biru dalam TBLA.

Lose Big Program (LBP) didesain sesuai format TBLA, yaitu mengombinasikan olah tubuh, serta sesi edukasi dan motivasi yang dilakukan berkelompok. Dalam program terstruktur yang berdurasi 13 minggu ini peserta dibagi dalam dua grup yang masing-masing beranggotakan 8 orang (tim merah dan tim biru). Setiap grup akan dilatih oleh dua trainer, dengan durasi 90 menit setiap sesi. Setiap minggu, akan dipilih pemenang yang berhasil menurunkan berat badan terbanyak.

Program ini memang hanya berlaku selama 13 minggu. Setelah itu peserta diharapkan sudah cukup percaya diri untuk meneruskan latihan dan menerapkan pola makan yang sehat sendiri di rumah. Nah, di situlah pentingnya latihan secara berkelompok. "Ketika semangat Anda mulai menurun, Anda akan selalu dimotivasi oleh teman-teman Anda. Sebab mereka punya tujuan yang sama, dan peduli dengan tujuan Anda itu," ujar Dave Nuku, dalam peluncuran program LBP di Fitness First Pacific Place, Kamis (18/3/2010) lalu.

Namun memang tak sembarang orang bisa mengikuti program ini. LBP ditujukan bagi orang yang mengalami obesitas dengan Indeks Massa Tubuh 25 atau lebih, serta persentase lemak tubuh 35 persen ke atas. Mereka inilah yang selama ini dianggap memiliki kesulitan menurunkan berat badan, dan sedang mencari solusinya.

LBP sejauh ini sudah dilakukan di Singapura dan Malaysia. Di Indonesia, program ini bisa diikuti di empat klub Fitness First, yaitu di Mal Taman Anggrek, Oakwood, Pacific Place, dan Senayan City.

Anda berminat mengikuti program ini? Silakan melakukan registrasi secara online di www.timetolosebig.com, atau www.fitnessfirst.co.id, atau hubungi:
Fitness First Taman Anggrek (021-560 9700)
Fitness First Platinum Senayan City (021-7278 1333)
Fitness First Platinum Pacific Place (021-5140 0525)
Fitness First Platinum Oakwood, Kuningan (021-2554 2333)

Orang tidak suka makan sayur itu biasa. Yang aneh mungkin orang yang tidak suka makan buah. Tak seperti sayur yang kadang-kadang terasa pahit atau hambar, buah memiliki rasa yang menyenangkan: asam, manis, atau kombinasi keduanya. Entah mengapa kedua jenis makanan ini sering tak disentuh. Padahal, bila Anda tidak rutin mengonsumsinya, Anda bisa mengalami konstipasi. Konstipasi hanya lah satu hal yang menunjukkan bahwa Anda kekurangan serat. Di luar itu, ada beberapa gejala lain yang menunjukkan bahwa Anda sebenarnya kekurangan serat. Bila Anda menyadari bahwa gejala ini terjadi pada Anda, segera perbaiki pola makan Anda.

* Sembelit. Jika dalam seminggu Anda hanya buang air besar (BAB) tiga kali, dan fesesnya terasa keras dan kering, artinya Anda memang mengalami sembelit, atau konstipasi. Konstipasi merupakan akibat kurangnya serat, jarang bergerak, dan selain itu karena beberapa pengobatan atau suplemen tertentu yang Anda konsumsi. Jika konstipasi ini berhubungan dengan apa yang Anda konsumsi, cobalah menambahkan lebih banyak makanan berserat, seperti apel, rasberi, wortel, brokoli, dan gandum utuh. Tambahkan perlahan-lahan sampai tubuh Anda terbiasa dengan asupan ini. Jangan lupa minum banyak air putih dan berolahraga secara teratur.

* Penambahan berat badan. "Serat itu memberikan rasa kenyang," ujar Kathleen Zelman, MPH, RD, direktur nutrisi WebMD. Rasa kenyang yang didapat juga terasa pas, tidak berlebihan. Jika Anda tidak merasakannya, berarti Anda makan lebih banyak daripada yang diperlukan oleh tubuh. Coba penuhi kebutuhan serat harian Anda sebanyak 25-35 gram, dengan menikmati makanan kaya serat seperti buah segar, gandum utuh, dan sayuran.

* Gula darah naik-turun. Jika Anda mengidap diabetes, dan sulit mengontrol kadar gula darah Anda, kemungkinan Anda kekurangan serat. Karena serat berfungsi menunda penyerapan gula, dan membantu Anda mengontrol kadar gula darah, coba tambahkan lebih banyak makanan segar, seperti kacang polong dan buncis, nasi merah, dan makanan berserat tinggi lainnya. Jangan lupa konsultasikan dulu dengan dokter.

* Mual akibat diet dan kelelahan. Kalori yang didapat hanya dari diet rendah karbohidrat dan tinggi protein (banyak daging, ) tidak hanya menyebabkan kolesterol naik, tetapi juga membuat Anda mual, lelah, dan lemah. Tambahkan gandum utuh yang mengandung banyak vitamin dan mineral, buah-buahan dan sayuran, dan kurangi makanan berlemak.

You might think there’s nothing special about this bridge, and Milliau Viaduct is indeed one of the more common-looking bridges, but the mere fact that it’s slightly taller than the Eiffel Tower makes it special.

Opened to traffic in December of 2004, Milliau Viaduct holds the current record for the world’s tallest vehicular bridge in the world, standing at an amazing 343 meters in its highest point, which makes it only 38 meters shorter than the Empire State Building. The bridge is set in Milliau, France is a part of the A75-A71 autorute from Paris to Beziers and it most likely lose its position as highest bridge deck in the world, when Chenab Bridge is completed in 209, in India, so we thought we’d mention it until then.

Living in Australia presents the following risk: mistake giant snakes for computer cables. Okay so you could spot it if your really careful but it blends in pretty well. So if one of your cables feels slippery and moving, take a good look at it.

You might think that if you come across a deer while going fishing in the ocean right? Well I guess that’s what these boys thought to themselves when they saw this dow while they were fishing quite a long way from shore in Chesapeake Bay. The fish weren’t biting that day so they had time to look around and spot something strange in the water. This deer was swimming for its life and it looked like it had been doing that for some time. When it saw the boat it started to swim towards it but when it saw the guys it kept its distance. One of them had some cowboy experience and using a lasso reeled it in the boat. It was so tired that it couldn’t even stay on its feet, plus it was scared as hell.

When they reached shore they tried to set it loose but it still couldn’t move so they left it there, but after a while she recovered her strength and moved on.

While religious experts search for proof Noah’s Arc really existed, modern artists build replicas of the ship from The Book of Genesis.

Johan Huibers a Dutch creationist from Schagen, Netherlands spent a lot of time and approximately $1.2 million to build a modern-day Noah’s Ark that’s 1/5 of the one described in The Bible. He started thinking about this project back in 1992 when the idea that his home country could become flooded due to global warming. Even when the idea became less popular he still continued to think about it and realized it was a great way to bring his fellow countrymen closer to religion.

So he began work on his 70 meters-long, 9,5 meters-wide and 13 meters-high arc in 2005 with money gathered from bank loans and now it’s ready to sail and calibrated to narrowly pass under every bridge in the interior waters of Netherlands. Although he couldn’t use the same materials described in The Book of Genesis, this modern Noah’s Arc is a nice replica and it even has some animal models to make it look more real.