I want to pave a 2 times n rectangle with blocks of two types, A and B, as illus

Question

I want to pave a 2 times n rectangle with blocks of two types, A and B, as illustrated at the right-long edges are length 2 and short edges arc length 1. I want to know in how many ways this can be done. To be more precise, at the right I have given three pavings for the ease n = 10. In fact the second diagram is a right-left reflection of the first and the third is an upside-down copy of the first, and the point I want to make is that they count as three "different" pavings. Find a linear recursive equation for x_n, the number of pavings of a 2 times n rectangle. [This is hard and requires an innovative idea.] Find enough initial conditions that Excel can be used to iterate the equation and find the number of pavings for n = 13.

Explanation / Answer

For 1a you could use induction:

Basically: Assume n=2 (at least two pizzas, otherwise no difference) and maximize difference. Let one of the pizzas have a diameter of 40 and the other one x Then we have 40-x < 40/(2-1) which is true.

Induction step n=>n+1 and you can try from there...

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