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Unformatted text preview: 6.042/18.062J Mathematics for Computer Science October 20, 2010 Tom Leighton and Marten van Dijk Notes for Recitation 12 1 The L-tower problem Observe the structures shown in Figure 1 . One has 2 L-shapes, the other 5 L-shapes. Con- sider a tower with k L-shapes. Assume that the blocks are all of size x × 1 where x > 1. As the picture indicates, if k is too small then the tower falls to the left. On the other hand, if k is too large the tower would fall to the right. Will the tower be stable for some k ? Prove there is at least one value of k for which the L-tower is stable. Assume that a structure is stable if and only if its center of gravity is not hanging in the air horizontally. The L-tower is stable if and only if each of its subparts is stable. Hint: Show the tower is stable if and only if 3 x 2 − 3 ≤ k ≤ 3 x 2 − 1 . Figure 1: Too few or too many L shapes make the tower unstable Solution. As indicated in the description, an arbitrary structure is considered stable if and only if its center of gravity lies on top of some form of support. For our L-towers this implies several conditions that must hold simultaneously. Consider the case k = 3, 2 Recitation 12 Condition 3 Condition 2 Condition 1 Figure 2: Three necessary conditions for stability, when k = 3 . Put together, we also consider them suﬃcient. illustrated in Figure 2 . For the structure to be stable, we need the topmost L-shape to be stable on top of the second L-shape. This is Condition 1. We also need the structure formed by the topmost 2 L-shapes to be overall stable on top of the lowest L-shape. This is Condition 2. And finally, we need the three L-shapes to be stable on top of the single standing block. Call this Condition 3. In general,...
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This note was uploaded on 01/19/2012 for the course CS 6.042J / 1 taught by Professor Tomleighton,dr.martenvandijk during the Fall '10 term at MIT.
- Fall '10
- Computer Science