Show all work. The Math for Computer Science mascot. Theory Hippotamus, made a s
ID: 2980656 • Letter: S
Question
Show all work.
The Math for Computer Science mascot. Theory Hippotamus, made a startling discovery while playing with his prized collection of unit squares over the weekend. Here is what happened. First, Theory Hippotamus put his favorite unit square down on the floor as in Figure 5.9 (a). He noted that the length of the periphery of the resulting shape was 4, an even number. Next, he put a second unit square down next to the first so that the two squares shared an edge as in Figure 5.9 (b). He noticed that the length of the periphery of the resulting shape was now 6, which is also an even number. (The periphery of each shape in the figure is indicated by a thicker line.) Theory Hippotamus continued to place squares so that each new square shared an edge with at least one previously-placed square and no squares overlapped. Eventually, he arrived at the shape in Figure 5.9 (c). He realized that the length of the periphery of this shape was 36, which is again an even number. Our plucky porcine pal is perplexed by this peculiar pattern. Use induction on the number of squares to prove that the length of the periphery is always even, no matter how many squares Theory Hippotamus places or how he arranges them.Explanation / Answer
P(n) = even number( where n is number of squares)
Base case ==> P(1) = 4 =even number
Assume P(k) = even
Now P(k+1) ,when the new square shares one edge , no. of peripheries = P(k) +4 (by new square) - 2(common edge) = even
when the new square shares two edges , no. of peripheries = P(k) +4 (by new square) - 4(2 common edges)=even
when the new square shares three edges , no. of peripheries = P(k) +4 (by new square) - 6(3 common edges)=even
when the new square shares three edges , no. of peripheries = P(k) +4 (by new square) -8(3 common edges)=even
Hence proved .
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