Ping-Pong Cells
I am an -dimensional creature working at a ping-pong ball factory. Today, I made a collection of ping-pong balls of the following form: the first was a 1-dimensional line with side length one, the second was a 2-dimensional triangle with side length two, the third was a 3-dimensional tetrahedron with side length three, and so on. I made figures up to an -cell with side length , then counted all the balls in all the figures. It turned out that the number of balls totaled a square integer! What is the smallest dimension that I could live in?
The answer is 4.
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1 solution
An n-cell is a figure in n-1 dimensions that connects n points, each in a new dimension, to each other. The formula for finding the total number of balls in an n-cell with side length x is n ! ( x ) ( x + 1 ) ( x + 2 ) ( x + 3 ) … ( x + n − 1 ) . The "line" will be made up of one ball, the triangle will be made up of 2 ( 2 ) ( 3 ) = 3 balls, and the tetrahedron will be made up of 3 ! ( 3 ) ( 4 ) ( 5 ) = 1 0 balls. The next figure, a 5-cell, will contain 4 ! ( 4 ) ( 5 ) ( 6 ) ( 7 ) = 3 5 balls. At this point, the total is 49, which is a perfect square. Thus, the answer is four.