# tetris tiling 2xn or ' filling a tube '

In how many ways can a rectangle $$2$$ units wide and $$8$$ units high be tiled with Tetris blocks? Extra credit: Let n be an even integer. In how many ways can a rectangle $$2$$ units wide and $$n$$ units high be tiled with Tetris blocks? I found a recursion that leads to $$F_{n-1}F_n$$ Fibonacci but still confused as to whether I found all configurations of each tetromino. ruling out the 'S' and 'T' piece we only need to find the number of ways the square, straight $$4$$, and L piece can fit into the $$2\times n$$ 'tube' right? someone always helps on here, thanks in advance.

• The S and T pieces can fit into the tube if oriented correctly, so why do you leavce them out? – Michael Lugo Dec 4 '18 at 16:26
• because an S piece will leave holes as well as the T pieces, try drawing it .a 2 square wide 'tube' , not the standard Tetris Tm game area. – Randin Dec 4 '18 at 17:19
• That makes sense. For others who don't see it: the area below an S or T piece is odd, but we need the area to be a multiple of 4 in order to fill it in. – Michael Lugo Dec 4 '18 at 20:39

## 1 Answer the proof is in the picture , there are a total of six different configurations

• Ross Millikan..where did u go buddy? 😑 – Randin Dec 4 '18 at 21:31
• But STILL my question is WHY for the B_n is there only two configurations and not 4 ? because both the shapes can be swapped / reflected meaning 2 more possabilities right ???!! – Randin Dec 5 '18 at 3:17
• achille hui can u draw what u mean ? that way i might better see what u are saying – Randin Dec 5 '18 at 9:12
• is the gray the shaded area ? and which configuration above cant have an L piece ? – Randin Dec 5 '18 at 9:35
• i meant what do u mean by gray area ? in my pic the gray areas represent already filled parts of the tube . u can also just flip the tubes over and then it is finding number of ways filling from the bottom which is the same as looking as top pieces and thier configurations – Randin Dec 10 '18 at 4:24