Perkembangan Facebook di Indonesia sungguh luar biasa. Berbagai kalangan mulai memanfaatkan situs generasi web 2.0 ini untuk berbagai hal. Mulai ajang untuk mengkampanyekan diri (caleg), menjual jasa/service, bertukar pikiran dalam komunitas, menjalin pertemanan sampai ajang menjaga 'nilai jual' keartisan.
Tercatat berbagai macam nama artis aktif menggunakan FB sebagai media untuk mendengar suara dari penggemar mereka dan berinteraksi. Sebutlah nama seperti Nafa Urbach dan komedian Parto Patrio. Mereka ini cukuf aktif menyapa dan berinteraksi dengan penggemarnya.
Nafa Urbach bahkan sering mendiskusikan berbagai jenis lagu sampai desain album barunya. Cukup menarik karena dalam proses ini Nafa sangat memperhatikan komentar dari fansnya. Sebuah proses marketing yang sangat mendengar suara konsumen utamanya.
Berbeda halnya dengan Parto Patrio. Di tengah kesibukan syuting jadwal padatnya, komedian satu ini sering menyapa para penggemarnya. Uniknya, Parto sampai menyempatkan diri menulis pantun bagi penggemarnya sampai membalas menulis di wall para fansnya hampir setiap hari.
Hal ini dilakukan lewat piranti BlackBerry yang digenggamnya. Berikut salah satu kutipan statusnya: 6 maret 2009, Parto : Liat bebek lagi kawin ama soang, ada orang lewat 'mereka pada kabur,,,, kerja semangat hari jum'at doang, apa karena besok semua pada libur. 10:05am-Comment-Like*
Tak dapat dipungkiri, selain menjaga awarness masyarakat akan dirinya, langkah cerdas ini akan menjaring penggemar baru bagi sang komedian. Sampai-sampai para penggemarnya mendirikan group POS FB (singkatan dari Perkumpulan Orang Sarap FB). Di dalamnya terjadi interaksi bercanda yang sangat ampuh untuk menghilangkan stress ataupun kepenatan sehari-hari (balas berbalas pantun dan becanda antara Parto dan para penggemarnya). Suatu langkah marketing yang bijak. Low budget high impact, what do you think?.
Piranti Blackberry yang digunakan oleh Parto, saat ini juga menuai manisnya booming dari facebook. BB (panggilan trend Blackberry) seolah menjadi piranti yang ampuh untuk melakukan posting di Facebook. Karena selain digunakan untuk email, kegiatan lain yang umum dilakukan pengguna BB adalah YM (Yahoo Messenger) dan Facebook.
Suatu momentum yang sangat menguntungkan penjualan Blackberry di Indonesia. Sampai-sampai 3 (Three) operator GSM di Indonesia, mencantumkan promosi internetnya yang dapat digunakan untuk facebook (catat: Facebook bukan Friendster) di iklan tvnya. Suatu pertanda untuk Friendster agar melakukan inovasi agar tidak tertinggal jauh oleh Facebook.
Para calon legislatif pun tidak segan-segan untuk mengucurkan dana kampanye untuk berpromosi di Facebook. Presiden SBY pun telah memiliki page di Facebook untuk menangkap trend web 2.0 dan mensosialisasikan diri di era kemajuan informasi ini. Termasuk Presiden Obama tentunya.
Nah sekarang, apakah anda akan tertarik untuk menyebarkan pesona anda guna menjaring pengguna Facebook di Indonesia sebanyak 1.964.580?
Minggu, 09 Mei 2010
Selasa, 04 Mei 2010
Autodesk® Robot™ Structural Analysis Professional 2010
Autodesk Robot Structural Analysis Professional 2010 software provides a scalable, country-specific analysis solution for the structural engineer to help analyze many types of structures, including buildings, bridges, civil and specialty structures. Autodesk Robot Structural Analysis Professional calculates a wide variety of structures with a comprehensive collection of design codes, delivering results in minutes, not hours. This structural engineering software is versatile enough for simple structural frame analysis or complex finite element analysis and offers smooth interoperability with Autodesk companion products as well as an open application programming interface (API).
What’s New
Enhanced Modeling and Collaboration Capabilities
* Integrate more efficiently between Autodesk Robot Structural Analysis Professional and Autodesk® Revit® Structure.
* New approach to building structures with the added ability to create objects such as beams columns, and walls in lieu of bars and panels.
* Import DXF™/DWG files as a drawing background, making it possible to define 3D structural models on the basis of architectural plans and elevations.
More Improved Design and Analysis Experience
* New harmonic and damping analysis parameters have been added, as well as a more improved and amended footfall analysis.
* New national reinforcement, load combination and steel codes have been implemented into the application, as well as regional codes for Sweden, Australia and Italy.
Greater Usability
* Enhanced creation of load distributions allows users to better analyze load distribution to beams, columns and walls.
* Work more easily with improved, user-friendly steel connections definitions, with updated dialog boxes.
* Work more easily with Eurocode safety factors with a comprehensive dialog box that contains all Eurocodes available in the program and an additional user-defined set of parameters.
Screenshots
Import DXF/DWG as backgrounds for defining 3D structural models on the basis of plans and elevations of structures received from the architect.
The predefined calculation models of slab, diaphragm, and deck decrease a number of degrees of freedom and eliminate vibration modes that are negligible for the seismic analysis (such as vibrations of slabs in their planes.
The trapezoidal and triangular method can distribute the load from the cladding to all objects located in the contour and plane of the cladding.
What’s New
Enhanced Modeling and Collaboration Capabilities
* Integrate more efficiently between Autodesk Robot Structural Analysis Professional and Autodesk® Revit® Structure.
* New approach to building structures with the added ability to create objects such as beams columns, and walls in lieu of bars and panels.
* Import DXF™/DWG files as a drawing background, making it possible to define 3D structural models on the basis of architectural plans and elevations.
More Improved Design and Analysis Experience
* New harmonic and damping analysis parameters have been added, as well as a more improved and amended footfall analysis.
* New national reinforcement, load combination and steel codes have been implemented into the application, as well as regional codes for Sweden, Australia and Italy.
Greater Usability
* Enhanced creation of load distributions allows users to better analyze load distribution to beams, columns and walls.
* Work more easily with improved, user-friendly steel connections definitions, with updated dialog boxes.
* Work more easily with Eurocode safety factors with a comprehensive dialog box that contains all Eurocodes available in the program and an additional user-defined set of parameters.
Screenshots
Import DXF/DWG as backgrounds for defining 3D structural models on the basis of plans and elevations of structures received from the architect.
The predefined calculation models of slab, diaphragm, and deck decrease a number of degrees of freedom and eliminate vibration modes that are negligible for the seismic analysis (such as vibrations of slabs in their planes.
The trapezoidal and triangular method can distribute the load from the cladding to all objects located in the contour and plane of the cladding.
Sejarah Sepak Bola
Asal muasal sejarah munculnya olahraga sepak bola masih mengundang perdebatan. Beberapa dokumen menjelaskan bahwa sepak bola lahir sejak masa Romawi, sebagian lagi menjelaskan sepak bola berasal dari tiongkok. FIFA sebagai badan sepak bola dunia secara resmi menyatakan bahwa sepak bola lahir dari daratan Cina yaitu berawal dari permainan masyarakat Cina abad ke-2 sampai dengan ke-3 SM. Olah raga ini saat itu dikenal dengan sebutan “tsu chu “.
Dalam salah satu dokumen militer menyebutkan, pada tahun 206 SM, pada masa pemerintahan Dinasti Tsin dan Han, masyarakat Cina telah memainkan bola yang disebut tsu chu. Tsu sendiri artinya “menerjang bola dengan kaki”. sedangkan chu, berarti “bola dari kulit dan ada isinya”. Permainan bola saat itu menggunakan bola yang terbuat dari kulit binatang, dengan aturan menendang dan menggiring dan memasukkanya ke sebuah jaring yang dibentangkan diantara dua tiang.
Versi sejarah kuno tentang sepak bola yang lain datangnya dari negeri Jepang, sejak abad ke-8, masyarakat disana telah mengenal permainan bola. Masyarakat disana menyebutnya dengan: Kemari. Sedangkan bola yang dipergunakan adalah kulit kijang namun ditengahnya sudah lubang dan berisi udara.
Menurut Bill Muray, salah seorang sejarahwan sepak bola, dalam bukunya The World Game: A History of Soccer, permainan sepak bola sudah dikenal sejak awal Masehi. Pada saat itu, masyarakat Mesir Kuno sudah mengenal teknik membawa dan menendang bola yang terbuat dari buntalan kain linen.
Sisi sejarah yang lain adalah di Yunani Purba juga mengenal sebuah permainan yang disebut episcuro, tidak lain adalah permainan menggunakan bola. Bukti sejarah ini tergambar pada relief-relief museum yang melukiskan anak muda memegang bola dan memainkannya dengan pahanya.
Sejarah sepak bola modern dan telah mendapat pengakuan dari berbagai pihak, asal muasalnya dari Inggris, yang dimainkan pada pertengahan abad ke-19 pada sekolah-sekolah. Tahun 1857 beridiri klub sepak bola pertama di dunia, yaitu: Sheffield Football Club. Klub ini adalah asosiasi sekolah yang menekuni permainan sepak bola.
Pada tahun 1863, berdiri asosiasi sepak bola Inggris, yang bernama Football Association (FA). Badan ini yang mengeluarkan peraturan permainan sepak bola, sehingga sepak bola menjadi lebih teratur, terorganisir, dan enak untuk dinikmati penonton.
Selanjutnya tahun 1886 terbentuk lagi badan yang mengeluarkan peraturan sepak bola modern se dunia, yaitu: International Football Association Board (IFAB). IFAB dibentuk oleh FA Inggris dengan Scottish Football Association, Football Association of Wales, dan Irish Football Association di Manchester, Inggris.
Sejarah sepak bola semakin teruji hingga saat ini IFAB merupakan badan yang mengeluarkan berbagai peraturan pada permainan sepak bola, baik tentang teknik permainan, syarat dan tugas wasit, bahkan sampai transfer perpindahan pemain.
Dalam salah satu dokumen militer menyebutkan, pada tahun 206 SM, pada masa pemerintahan Dinasti Tsin dan Han, masyarakat Cina telah memainkan bola yang disebut tsu chu. Tsu sendiri artinya “menerjang bola dengan kaki”. sedangkan chu, berarti “bola dari kulit dan ada isinya”. Permainan bola saat itu menggunakan bola yang terbuat dari kulit binatang, dengan aturan menendang dan menggiring dan memasukkanya ke sebuah jaring yang dibentangkan diantara dua tiang.
Versi sejarah kuno tentang sepak bola yang lain datangnya dari negeri Jepang, sejak abad ke-8, masyarakat disana telah mengenal permainan bola. Masyarakat disana menyebutnya dengan: Kemari. Sedangkan bola yang dipergunakan adalah kulit kijang namun ditengahnya sudah lubang dan berisi udara.
Menurut Bill Muray, salah seorang sejarahwan sepak bola, dalam bukunya The World Game: A History of Soccer, permainan sepak bola sudah dikenal sejak awal Masehi. Pada saat itu, masyarakat Mesir Kuno sudah mengenal teknik membawa dan menendang bola yang terbuat dari buntalan kain linen.
Sisi sejarah yang lain adalah di Yunani Purba juga mengenal sebuah permainan yang disebut episcuro, tidak lain adalah permainan menggunakan bola. Bukti sejarah ini tergambar pada relief-relief museum yang melukiskan anak muda memegang bola dan memainkannya dengan pahanya.
Sejarah sepak bola modern dan telah mendapat pengakuan dari berbagai pihak, asal muasalnya dari Inggris, yang dimainkan pada pertengahan abad ke-19 pada sekolah-sekolah. Tahun 1857 beridiri klub sepak bola pertama di dunia, yaitu: Sheffield Football Club. Klub ini adalah asosiasi sekolah yang menekuni permainan sepak bola.
Pada tahun 1863, berdiri asosiasi sepak bola Inggris, yang bernama Football Association (FA). Badan ini yang mengeluarkan peraturan permainan sepak bola, sehingga sepak bola menjadi lebih teratur, terorganisir, dan enak untuk dinikmati penonton.
Selanjutnya tahun 1886 terbentuk lagi badan yang mengeluarkan peraturan sepak bola modern se dunia, yaitu: International Football Association Board (IFAB). IFAB dibentuk oleh FA Inggris dengan Scottish Football Association, Football Association of Wales, dan Irish Football Association di Manchester, Inggris.
Sejarah sepak bola semakin teruji hingga saat ini IFAB merupakan badan yang mengeluarkan berbagai peraturan pada permainan sepak bola, baik tentang teknik permainan, syarat dan tugas wasit, bahkan sampai transfer perpindahan pemain.
Graphs Fundamentals
Several puzzles on these pages (Sam Loyd's Fifteen, Sliders, Lucky 7, Happy 8, Blithe 12) could be better understood with the help of the Graph Theory. While it does not immediately offer all the answers it does provide a unified and illuminating approach to these and many other puzzles and games. The Graph Theory originates with a 1736 Leonard Euler's paper "The Seven Bridges of Königsberg". This is how Euler describes the problem:
The problem, which I understand is well known, is stated as follows:
In the town of Königsberg in Prussia (nowadays a city of Kaliningrad in Russia, comment is mine, CTK) there is an island A, called "Kneiphoff", with the two branches of the river (Pregel) flowing around it. There are seven bridges, a, b, c, d, e, f, and g, crossing the two branches.The question is whether a person can plan a walk in such a way that he will cross each of these bridges once but not more than once. I was told that while some deny the possibility of doing this and others were in doubt, there were none who maintained that it was actually possible.On the basis of the above I formulated the following very general problem for myself: Given any configuration of the river and the branches into which it may divide, as well as any number of bridges, to determine whether or not it is possible to cross each bridge exactly once.
I want to sketch a short proof of impossibility to construct such a pass using Graph Theory terms which will be introduced later on. You may skip it and return here later, or you may read through and, perhaps, gain an independent insight into the power of generalization afforded by the Graph Theory.
1. The sum of degrees of the vertices of a graph is even
2. Every graph has an even number of odd vertices
1. If the number of odd vertices is greater than 2 no euler walk exists
2. If the number of odd vertices is 2, euler walks exist starting at either of the odd vertices
3. With no odd vertices, euler walks can start at an arbitrary vertex
Euler abstracted the bridges into edges and pieces of land into nodes of a graph.
A graph is a collection of nodes (also called vertices) and edges (also called links) each connecting a pair of nodes.
As far as the definitions go, this one is not very good. It's ambiguous for at least two reasons. This definition depends on notions that have not been defined: collection and connecting.
1.
The notion of collection is too fundamental to be treated here. It is indeed customary (under this pretext) to dodge defining a collection by simply appealing to one's common sense (like in, "Collection is a set, aggregation of objects possessing a certain attribute that separates them from all other objects as members of the given set."). I believe everyone agrees that as long as this notion is being used fleetingly as a tool (much like a pencil or a computer) it's OK not to go into details which would definitely divert us from the topic at hand.
But note also that two nodes that define an edge could coalesce into one or can be joined by more than one edge. Thus the "collection" in question is not necessarily a set, but rather a multiset.
2.
The second notion, that of the edges being connections between nodes, is by far too important to the Graph Theory to leave it to one's intuitive perception. It's worse still to conceal the ambiguity of the definition by slipping in a graphic diagram with an immediate appeal to one's intuition. Are the graphs on the two diagrams below the same or not?
Thus, to be more specific, an edge is a 1- or 2-element set with elements drawn from the vertex set. An edge is said to connect the vertices which are its elements. A 1-element edge, i.e. an edge whose ends coincide, is called a loop.
For the visualization sake I shall use the diagrams. I believe it's also OK in so far as they are perceived critically. For example, the two graphs on the diagrams above are of course the same.
Definitions
1. Two vertices of a graph are adjacent if they belong to the same edge.
2. Elements of an edge are said to be incident to that edge.
3. Likewise, an edge is incident to its elements.
4. A degree of a vertex is the number of edges incident to it (loops being counted twice).
5. A vertex of odd (even) degree is said to be odd (even).
Proposition (the Handshake Lemma)
For a graph, the sum of degrees of all its nodes equals twice the number of edges.
Proof
A degree of a node is the number of edges incident to this node. However every edge is incident to two nodes (or, for a loop, twice to the same node).
Corollary
For a graph, the sum of degrees of all its nodes is even.
The applet below is intended to help you with the just introduced concepts. To create a link drag the mouse from one node to another. You may press or release the button either on an existent node or anywhere else. In the latter case, a new node will be created. If the box "Move vertex" is checked you will drag the vertices around instead. The multiplicity of a link is shown by an integer in the middle of the link.
This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at http://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.
Buy this applet
What if applet does not run?
The applet may prompt one other observation: the number of odd nodes is always even. Indeed, the sum of all node degrees is even. The sum of the degrees of the even nodes is naturally even. Subtracting one from the other we see that the sum of the degrees of the odd nodes is also even.
M. Gardner quotes a proof by Gerald K. Schoenfeld, a medical officer in the U. S. Navy. Let the nodes denote the participants in a medical convention. The edges represent the handshakes between two participants. Starting at the opening of the convention, as the participants greet each other with shakehands, the edges are being added. At the outset, the number of edges was 0, an even number. A participant is designated odd or even depending on the parity of the number of handshakes he/she took part in up to the given moment. A handshake may occur between three kinds of pairs: odd/odd, even/even, and odd/even. After the handshake, an odd/odd pair becomes even/even, thus the number of odd participants is reduced by 2. An even/even pair becomes odd/odd so that the number of odd participants is increased by 2. The odd/even pair becomes even/odd which does not affect the number of odd participants. We see that a handshake does not change the parity of the number of odd participants. Since at the outset it was even, it will remain even after any number of handshakes.
Definition
1. A walk (or a path) of length n on a graph is a sequence v0, e1, v1, e2, ..., vn, where vi are vertices while ei are edges of the graph such that vertices and edges adjacent in the sequence are incident.
2. A walk v0, e1, v1, e2, ..., vn is said to connect v0 and vn.
3. A walk is closed if v0n. A closed walk is called a cycle.
4. A walk which is not closed is open.
5. A walk is an euler walk if every edge of the graph appears in the walk exactly once.
6. A graph is connected if every two vertices can be connected by a walk.
Proposition
1. If a graph has a closed euler walk then every vertex is even.
2. If every vertex of a connected graph is even, the graph has an euler walk.
3. If a graph has an open euler walk it has exactly two odd vertices.
4. If a connected graph has exactly two odd vertices it also has an open euler walk.
Reference
1. S. Barr, Experiments in Topology, Dover Publications, 1964
2. A. Beck, M.N. Bleicher, D. W. Crowe. Excursions into Mathematics, A K Peters, 2000
3. J. L. Casti, Five Golden Rules, John Wiley & Sons, 1996
4. S. K. Stein, Mathematics: The Man Made Universe, Dover (January 26, 1999)
5. M. Gardner, The Colossal Book of Short Puzzles and Problems, W. W. Norton, 2006, p. 27
6. S.K.Stein, Mathematics: The Man-Made Universe, 3rd edition, Dover, 2000.
7. R.J.Trudeau, Introduction to Graph Theory, Dover, NY, 1993.
On Internet
1. Combinatorial Object Server
An aside
Königsberg is the birth place of the famous German philosopher Immanuel Kant (1724-1804). I have never visited the place which Kant is known to have never left. Russia annexed this piece of Germany after the World War II. I often wondered if Kant should be considered a great Russian philosopher. After the Baltic states gained their independence from Russia a few years back, the region has no direct link to Russia at all.
* Graphs Fundamentals
o Crossing Number of a Graph
o Regular Polyhedra
o 3 Utilities Puzzle: Water, Gas, Electricity
o A Clique of Acquaintances
o Round Robin Tournament
o The Affirmative Action Problem
o The Two Men of Tibet Problem
* Puzzles on graphs
o Three Glass Puzzle (example)
o Three Glass Puzzle in Barycentric Coordinates (example)
o Sierpinski Gasket and Tower of Hanoi (example)
o Sam Loyd's Fifteen
o Sliders
o Lucky 7
o Happy 8
o Blithe 12
o Slider puzzles
* Permutations
o Various ways to define a permutation
o Counting and listing all permutations
* Transpositions
o A Shuttle Puzzle
* Groups of Permutations
o Group multiplication of permutations
The problem, which I understand is well known, is stated as follows:
In the town of Königsberg in Prussia (nowadays a city of Kaliningrad in Russia, comment is mine, CTK) there is an island A, called "Kneiphoff", with the two branches of the river (Pregel) flowing around it. There are seven bridges, a, b, c, d, e, f, and g, crossing the two branches.The question is whether a person can plan a walk in such a way that he will cross each of these bridges once but not more than once. I was told that while some deny the possibility of doing this and others were in doubt, there were none who maintained that it was actually possible.On the basis of the above I formulated the following very general problem for myself: Given any configuration of the river and the branches into which it may divide, as well as any number of bridges, to determine whether or not it is possible to cross each bridge exactly once.
I want to sketch a short proof of impossibility to construct such a pass using Graph Theory terms which will be introduced later on. You may skip it and return here later, or you may read through and, perhaps, gain an independent insight into the power of generalization afforded by the Graph Theory.
1. The sum of degrees of the vertices of a graph is even
2. Every graph has an even number of odd vertices
1. If the number of odd vertices is greater than 2 no euler walk exists
2. If the number of odd vertices is 2, euler walks exist starting at either of the odd vertices
3. With no odd vertices, euler walks can start at an arbitrary vertex
Euler abstracted the bridges into edges and pieces of land into nodes of a graph.
A graph is a collection of nodes (also called vertices) and edges (also called links) each connecting a pair of nodes.
As far as the definitions go, this one is not very good. It's ambiguous for at least two reasons. This definition depends on notions that have not been defined: collection and connecting.
1.
The notion of collection is too fundamental to be treated here. It is indeed customary (under this pretext) to dodge defining a collection by simply appealing to one's common sense (like in, "Collection is a set, aggregation of objects possessing a certain attribute that separates them from all other objects as members of the given set."). I believe everyone agrees that as long as this notion is being used fleetingly as a tool (much like a pencil or a computer) it's OK not to go into details which would definitely divert us from the topic at hand.
But note also that two nodes that define an edge could coalesce into one or can be joined by more than one edge. Thus the "collection" in question is not necessarily a set, but rather a multiset.
2.
The second notion, that of the edges being connections between nodes, is by far too important to the Graph Theory to leave it to one's intuitive perception. It's worse still to conceal the ambiguity of the definition by slipping in a graphic diagram with an immediate appeal to one's intuition. Are the graphs on the two diagrams below the same or not?
Thus, to be more specific, an edge is a 1- or 2-element set with elements drawn from the vertex set. An edge is said to connect the vertices which are its elements. A 1-element edge, i.e. an edge whose ends coincide, is called a loop.
For the visualization sake I shall use the diagrams. I believe it's also OK in so far as they are perceived critically. For example, the two graphs on the diagrams above are of course the same.
Definitions
1. Two vertices of a graph are adjacent if they belong to the same edge.
2. Elements of an edge are said to be incident to that edge.
3. Likewise, an edge is incident to its elements.
4. A degree of a vertex is the number of edges incident to it (loops being counted twice).
5. A vertex of odd (even) degree is said to be odd (even).
Proposition (the Handshake Lemma)
For a graph, the sum of degrees of all its nodes equals twice the number of edges.
Proof
A degree of a node is the number of edges incident to this node. However every edge is incident to two nodes (or, for a loop, twice to the same node).
Corollary
For a graph, the sum of degrees of all its nodes is even.
The applet below is intended to help you with the just introduced concepts. To create a link drag the mouse from one node to another. You may press or release the button either on an existent node or anywhere else. In the latter case, a new node will be created. If the box "Move vertex" is checked you will drag the vertices around instead. The multiplicity of a link is shown by an integer in the middle of the link.
This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at http://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.
Buy this applet
What if applet does not run?
The applet may prompt one other observation: the number of odd nodes is always even. Indeed, the sum of all node degrees is even. The sum of the degrees of the even nodes is naturally even. Subtracting one from the other we see that the sum of the degrees of the odd nodes is also even.
M. Gardner quotes a proof by Gerald K. Schoenfeld, a medical officer in the U. S. Navy. Let the nodes denote the participants in a medical convention. The edges represent the handshakes between two participants. Starting at the opening of the convention, as the participants greet each other with shakehands, the edges are being added. At the outset, the number of edges was 0, an even number. A participant is designated odd or even depending on the parity of the number of handshakes he/she took part in up to the given moment. A handshake may occur between three kinds of pairs: odd/odd, even/even, and odd/even. After the handshake, an odd/odd pair becomes even/even, thus the number of odd participants is reduced by 2. An even/even pair becomes odd/odd so that the number of odd participants is increased by 2. The odd/even pair becomes even/odd which does not affect the number of odd participants. We see that a handshake does not change the parity of the number of odd participants. Since at the outset it was even, it will remain even after any number of handshakes.
Definition
1. A walk (or a path) of length n on a graph is a sequence v0, e1, v1, e2, ..., vn, where vi are vertices while ei are edges of the graph such that vertices and edges adjacent in the sequence are incident.
2. A walk v0, e1, v1, e2, ..., vn is said to connect v0 and vn.
3. A walk is closed if v0n. A closed walk is called a cycle.
4. A walk which is not closed is open.
5. A walk is an euler walk if every edge of the graph appears in the walk exactly once.
6. A graph is connected if every two vertices can be connected by a walk.
Proposition
1. If a graph has a closed euler walk then every vertex is even.
2. If every vertex of a connected graph is even, the graph has an euler walk.
3. If a graph has an open euler walk it has exactly two odd vertices.
4. If a connected graph has exactly two odd vertices it also has an open euler walk.
Reference
1. S. Barr, Experiments in Topology, Dover Publications, 1964
2. A. Beck, M.N. Bleicher, D. W. Crowe. Excursions into Mathematics, A K Peters, 2000
3. J. L. Casti, Five Golden Rules, John Wiley & Sons, 1996
4. S. K. Stein, Mathematics: The Man Made Universe, Dover (January 26, 1999)
5. M. Gardner, The Colossal Book of Short Puzzles and Problems, W. W. Norton, 2006, p. 27
6. S.K.Stein, Mathematics: The Man-Made Universe, 3rd edition, Dover, 2000.
7. R.J.Trudeau, Introduction to Graph Theory, Dover, NY, 1993.
On Internet
1. Combinatorial Object Server
An aside
Königsberg is the birth place of the famous German philosopher Immanuel Kant (1724-1804). I have never visited the place which Kant is known to have never left. Russia annexed this piece of Germany after the World War II. I often wondered if Kant should be considered a great Russian philosopher. After the Baltic states gained their independence from Russia a few years back, the region has no direct link to Russia at all.
* Graphs Fundamentals
o Crossing Number of a Graph
o Regular Polyhedra
o 3 Utilities Puzzle: Water, Gas, Electricity
o A Clique of Acquaintances
o Round Robin Tournament
o The Affirmative Action Problem
o The Two Men of Tibet Problem
* Puzzles on graphs
o Three Glass Puzzle (example)
o Three Glass Puzzle in Barycentric Coordinates (example)
o Sierpinski Gasket and Tower of Hanoi (example)
o Sam Loyd's Fifteen
o Sliders
o Lucky 7
o Happy 8
o Blithe 12
o Slider puzzles
* Permutations
o Various ways to define a permutation
o Counting and listing all permutations
* Transpositions
o A Shuttle Puzzle
* Groups of Permutations
o Group multiplication of permutations
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