Trouble with Dots
You are given 8 Points. You are told to create as many X shapes (With a point at each vertex) as possible with those 8 dots. You can rearrange them in any pattern you like. The X's may not cross over another point and the angle created by the lines of the x must be greater than 30 degrees.
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1 solution
There are two ways to solve this problem. The first way is more analytical and logical. The second is more mathematical.
First:
You arrange the eight dots in a row and label them with a letter each (A-H). You then create combinations of each that contain no repeated starting letter from combinations.
The A's: ABCD ACEF ABEG ACGH AEDH ABFH
The B's: BHGD BGCF BDEF
The C's: CDEG CFDH
The E's EFGH
And we can see that there are 12 solutions with four points each. Any other combination would result in a repeated A,B,C, or E.
Second: You look at the 8 points and rearrange them into a 3 dimensional cube. An X is created at any intersection of 4 points and there are exactly 12 intersections possible if you use every possible angle. You can use the first method to check the second method.
Please rate this question and this solution. I came up with this question in the middle of my calculus class while talking about the General Power rule. It took my 2 minutes to solve it completely, how long did it take you?
Thanks for reading.
~David Bloom