<p>Is there any way to do this algebraically or at least systematically?</p>
<p>If you don't know what I mean, here's a sample question with the closest wording I can come up with.</p>
<p>What is the maximum number of intersections that 10 lines can have?</p>
<p>Isn’t it just 1+2+3+4+5+6+7+8+9?
Each line can intersect all of the previous ones a single time.</p>
<p>Just add up (n-1) to the n value of the number of lines, if I remember how to do these correctly.</p>
<p>Here is a logical way that I learned how to do this problem that can apply for any problems of this type: Once you learn this, it will be very quick for you, but here is an explanation of what I do:</p>
<p>You start off with 2 lines:</p>
<p>Logically, they can only intersect once. making them an X if you wish. Therefore: 1 intersection</p>
<p>Then you move to 3 lines:</p>
<p>Starting from the intersection of the two lines, the next line can intersect both lines once: This gives it three intersections.</p>
<p>If you add another line, starting from the 3 line example, you will quickly see the next line can intersect the other 3 lines once.</p>
<p>What you get is 6 intersections. You should quickly notice by now that the pattern is starting to develop:</p>
<p>2 lines= 1
3 lines= 1+2
4 lines= 1+2+3
..10 lines = 1+2+3+4+5+6+7+8+9</p>
<p>like the above posters have said…start from 1 and add up to (n-1) which is nine and you have your answer.</p>
<p>@khoitrinh: That makes sense to me logically, but I can’t seem to prove it by drawing it out myself. Anyone sure of the way?</p>
<p>Use a big piece of paper and thin pencil lead.</p>
<p>ya what everyone is saying is correct</p>
<p>The smaller the angle of intersection of the lines, the easier it will be to draw 10 lines that all intersect each other.</p>
<p>That formula is incorrect. That multiples all the numbers from n to (n-1). We want to add them together.</p>
<p>If you want to get into formulas, the correct one is [n(n-1)]/2.</p>
<p>A spent a couple of minutes in powerpoint drawing this for you. I accidentally did 11 lines, but I didn’t feel like redoing it. This is a good way to see how each line actually goes through every other one.</p>
<p>You can also prove it by using any 10 random linear equations. As long as they all have different slopes, you can calculate all the points of intersection between them.
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<p>I take it you had impressive Linerider skills back in the day.</p>
<p>Thanks everyone. Makes sense to me.</p>
<p>And by the way, the question can’t possibly be asking for infinity, right, like with all the lines on top of each other?</p>
<p>Yes the question would never ask for the lines that are on top of each other. Generally, the question will say something like “ten different lines” and lines on top of each other are the same line.</p>
<p>wow i never knew this</p>
<p>Wow thats one useful formula.</p>
