Growing Ls

9506 /www/nrich/html/content/id/9506/ 1 2012-09-25T13:26:17 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> This is a size 1 L: <div style="text-align: center;"><mdo:image alt="" height="112" src="trisquare.png" width="118"></mdo:image></div> Four of them can be used to tile a size 2 L: <div style="text-align: center;"><mdo:image alt="size 2 triomino tiled" height="212" src="Lshapes2.png" width="209"></mdo:image></div> Devise a convincing argument that you will be able to tile a size 4, 8, 16, 32... $2^n$ L using size 1 Ls. If you liked this question, <a href="/7026">here is an NRICH task</a> which challenges you to use similar mathematical ideas. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> First thing to note is that as 2 can be tiled, so can all powers of 2. By the same reasoning, if size n can be tiled, so can 2n. So we only need to be able to show that all odd (well, all primes really) can be tiled, and we're done. 3 can be tiled: <mdo:image height="150" width="150" src="3tiled.png" alt=""></mdo:image> from 3 we can tile 5, from 5 we can tile 7 and from 7 we can tile 9 (only half shown because of the symmetry) <mdo:image height="191" width="300" alt="" src="3to5.png"></mdo:image><mdo:image height="225" width="300" alt="" src="5to7.png"></mdo:image><mdo:image height="238" width="400" alt="" src="7to9.png"></mdo:image> To prove it can always be done, we just need to prove that for any odd n, the n+2th L can be tiled: <mdo:image height="226" width="300" src="proof.png" alt=""></mdo:image> This means filling the inverted L with the dimensions given. 3 different cases: when n is congruent to 0, 1 or 2 mod 3. When n congruent to 0, the L can be split into 3 n by 2 rectangles, which can be tiled (see 3 to 5 above) When n is congruent to 1, n+2 is congruent to 3, so the L can be split into a 4 by (n+2) rectangle (tiled with 3 by 2 rectangles) and an (n-4) by 2 rectangle - (n-4) is congruent to 0 mod 3 so this can be tiled. When n is congruent to 2, we split the shape into an (n-2) by 2 rectangle (can be tiled with 3 by 2 rectangles), an (n-2) by 4 rectangle (again can by tiled with 3 by 2 rectangles) and a size 2 L, which we know we can tile. </mdoxml> 2 4 0 0 1 0 0 Growing Ls Can you fit Ls together to make larger versions of themselves? Admin Short problems Secondary Mapping Document Enlargements and Scale factors Using, Applying and Reasoning about Mathematics Mathematical reasoning & proof