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But what does she mean? She is trying to say that she is not the person who said the man stole the money. Somebody else said it.

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You think I said it but I did not. What does the speaker mean now? It sounds like she wanted to suggest that the man stole the money. But she did not want to directly say it. Now for the next one. By now you might be able to guess the meaning.

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Listen here:. Here, the speaker is suggesting that someone else stole the money, not the man identified in the sentence. In this case, the speaker is suggesting that she is talking about some other money, not the specific money being discussed. Here, the speaker is suggesting that the man stole something else. For example, maybe he stole jewelry or some other valuables. At home, you can practice saying the sentence seven times, moving the stress to a different word each time.

Eight ways the world is not designed for women

Some of you may feel strange about putting stress on one specific word. But it is a communication tool that sounds perfectly natural in English when used correctly. As you pay attention to native English speakers, you will notice that we use the tool often. You can find examples on television and in films, for example.

See How to Edit for help, or this article's talk page. Seven is the most powerful magical number [1] , based on centuries of mythology, science , and mathematics, and therefore has a very important role in the wizarding world. Arithmancer Bridget Wenlock was the first to note this through a theorem which exposed the magical properties of the number seven. The Arithmancer , Bridget Wenlock — was the first witch to establish the magical properties of the number seven.

Euler proved that the problem has no solution. The difficulty he faced was the development of a suitable technique of analysis, and of subsequent tests that established this assertion with mathematical rigor. First, Euler pointed out that the choice of route inside each land mass is irrelevant. The only important feature of a route is the sequence of bridges crossed. This allowed him to reformulate the problem in abstract terms laying the foundations of graph theory , eliminating all features except the list of land masses and the bridges connecting them.

In modern terms, one replaces each land mass with an abstract " vertex " or node, and each bridge with an abstract connection, an " edge ", which only serves to record which pair of vertices land masses is connected by that bridge. The resulting mathematical structure is called a graph.

Since only the connection information is relevant, the shape of pictorial representations of a graph may be distorted in any way, without changing the graph itself. Only the existence or absence of an edge between each pair of nodes is significant. For example, it does not matter whether the edges drawn are straight or curved, or whether one node is to the left or right of another.

Next, Euler observed that except at the endpoints of the walk , whenever one enters a vertex by a bridge, one leaves the vertex by a bridge. In other words, during any walk in the graph, the number of times one enters a non-terminal vertex equals the number of times one leaves it.

Now, if every bridge has been traversed exactly once, it follows that, for each land mass except for the ones chosen for the start and finish , the number of bridges touching that land mass must be even half of them, in the particular traversal, will be traversed "toward" the landmass; the other half, "away" from it.

However, all four of the land masses in the original problem are touched by an odd number of bridges one is touched by 5 bridges, and each of the other three is touched by 3. Since, at most, two land masses can serve as the endpoints of a walk, the proposition of a walk traversing each bridge once leads to a contradiction.

In modern language, Euler shows that the possibility of a walk through a graph, traversing each edge exactly once, depends on the degrees of the nodes. The degree of a node is the number of edges touching it. Euler's argument shows that a necessary condition for the walk of the desired form is that the graph be connected and have exactly zero or two nodes of odd degree. This condition turns out also to be sufficient—a result stated by Euler and later proved by Carl Hierholzer. Such a walk is now called an Eulerian path or Euler walk in his honor.

Further, if there are nodes of odd degree, then any Eulerian path will start at one of them and end at the other.

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An alternative form of the problem asks for a path that traverses all bridges and also has the same starting and ending point. Such a walk is called an Eulerian circuit or an Euler tour. Such a circuit exists if, and only if, the graph is connected, and there are no nodes of odd degree at all. All Eulerian circuits are also Eulerian paths, but not all Eulerian paths are Eulerian circuits. Euler's work was presented to the St.

Petersburg Academy on 26 August , and published as Solutio problematis ad geometriam situs pertinentis The solution of a problem relating to the geometry of position in the journal Commentarii academiae scientiarum Petropolitanae in Combinatorial problems of other types had been considered since antiquity. In addition, Euler's recognition that the key information was the number of bridges and the list of their endpoints rather than their exact positions presaged the development of topology.

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The difference between the actual layout and the graph schematic is a good example of the idea that topology is not concerned with the rigid shape of objects. Hence, as Euler recognized, the "geometry of position" is not about "measurements and calculations" but about something more general.