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Tiling and counting (Posted on 2015-11-10) Difficulty: 2 of 5
In how many ways can a 2xn grid be tiled using only 2x1 tiles, which can be laid either horizontally or vertically?

Design a way to enumerate those patterns.

See The Solution Submitted by Ady TZIDON    
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Solution Solution and my story. Comment 2 of 2 |
The reason for the Fibonacci is that for a given n, you can take each tiling for n-2 and add = to each of them and then take each tiling for n-1 and add | to each of them.
Since there is 1 way to tile a 0xn and 1 way to tile a 1xn we have Fibonacci.

Years ago at a week-long conference on problem solving for math teachers.  I was trying to tackle a problem similar to this one but much more difficult.  Counting tilings of x tetrominoes in a nxm grid.  It came down to a recursion but I couldn't show why.

After a couple of days the class shared where we were with our 'hard' problems.  Later that day he had the whole class work on the problem here.  When I solved it and could show the recursion I immediately realized how amazing the techer was!  By the next day I had full formulas both recursive and explicit and I knew a lot more about how to teach problem solving to my own students.

  Posted by Jer on 2015-11-10 13:37:39
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