## 2 Construction  
The proof of Theorem 1 is done by a construction of the desired interlocked polygons for any monotone Boolean function \(f\) on \(n\) variables \(x_{1},\ldots,x_{n}\). The outline of the construction is depicted in Figure 3. The polygons \(B\), \(C\), \(D\), \(E\), and \(F\) form the outer frame. When polygon \(A\) is at the initial position as in the figure, they are interlocked. However, once we slide \(A\) left, \(B\) can be plugged out, and all of \(C\), \(D\) and \(E\) can be removed in this order, and then all polygons can be moved from inside of \(F\) to its outside, and they will be free. Hence we will construct the gadgets inside of this \"frame\". Each of the variables \(x_{1},\ldots,x_{n}\) corresponds to a polygon as depicted in Figure 3. If we remove the polygon \(x_{i}\) (we sometimes identify a variable/operator and the corresponding gadget) since \(x_{i}=1\), the input spread to right side. We note that each of \"input\" _pulls_ the polygons one unit length that may be different from typical reductions. Figure 3: Outline of the construction. In this frame, to complete the construction of the gadgets that realize a monotone function \(f\), we have to design five kinds of gadgets; (1) WIRE for wire to transfer a signal; (2) AND for \"and\" operation between two variables; (3) OR for \"or\" operation between two variables; (4) SPLIT to split a signal to two or more signals; (5) TURN to turn a signal; and (6) CROSS to cross two independent signals. Among them, WIRE, AND, and SPLIT are relatively simple to realize. In Figure 4(a), the operation (\(x_{1}\) and \(x_{2}\)) is computed by the AND block, and the signal is sent to right by the wire blocks, and split by the SPLIT block. (In fact, SPLIT is essentially the same as AND, which is used backward way.) In each move, each block moves one unit length. One may wonder what happens if some blocks are moved out from our assumption. For example, the wire block between \(x_{1}\) and AND in Figure 4(a) can be moved to up after removing \(x_{1}\). But we can avoid these unexpected cases; the blocks can be locked locally if we attach some long arms as shown in Figure 4(b). To simplify, we omit the arms in the other figures. The OR gadget is designed as in Figure 5. The initial position is depicted in Figure 5(a). If one of \(x_{1}\) and \(x_{2}\) moves a unit length far from the block OR, OR can be moved to left as in Figure 5(b). (In the figure, \(x_{1}\) moves to up.) We note that even if both \(x_{1}\) and \(x_{2}\) move a unit length far from the block OR, OR is still locked in with the blocks \(L_{1}\) and \(L_{2}\). Otherwise, moving just \(x_{1}\) may have some bad influence to the different variable \(x_{2}\) and vice versa; intuitively, this mechanism prevents to bounce back the signal of \(x_{1}\) to \(x_{2}\) through this gadget. The TURN gadget is designed as in Figure 6. The initial position is depicted in Figure 6(a). When the left block moves to left, the blocks can move as in Figure 5(b). Figure 4: The gadgets for WIRE, AND, and SPLIT. In many reductions, the crossover of the wires is the key gadget (see [2] for the details). But in this problem, the crossover can be realized by a simple gadget in Figure 7.  
Figure 5: OR gadget. Figure 6: TURN gadget. Figure 7: CROSS gadget. Using the gadgets, we can complete the construction of the polygons that computes \(f(x_{1},\ldots,x_{n})\), and the final output is connected to the polygon \(A\) in Figure 3.0 A simple example of the construction for \(f(x_{1},x_{2},x_{3})=((x_{1}\wedge x_{2})\lor x_{3})\wedge(x_{1}\lor x_{3})\) is shown in Figure 3.1 (We note that this example is illustrative, though it can be simplified logically.) Some blocks are stretched in a trivial way to adjust their size. Proof. : (of Theorem 1.) For any given monotone Boolean function \(f\) on \(n\) variables \(x_{1},\ldots,x_{n}\), we can construct the interlocked polygons \(S\) as described above.