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Bullpens Reflection – Craig Johnson The high mountain fencing company is also in the business of building bull pens. However pens to hold more than one bull are constructed differently than cow pens. Stage 1 – contains 1 bull Stage 2 – contains 2 bulls Stage 3 – contains 3 bulls I started off the problem by looking at the given figures, counting the squares and making a t-chart from the first three stages. By the time I had completed writing the 2 nd stage, I had made a mental conjecture for the 3rd guessing at a linear equation. I confirmed this by the completion of the t-chart for that level with the actual counting of squares from the example. Stage Number 1 2 3 4 5 10 100 n # of bulls 1 2 3 4 5 10 100 n Number of walls 8 13 18 23 23 53 503 5n + 3 Looking at the formula I realized a geometric explanation would be that each bull is surrounded by 5 walls and there are always 3 walls at the end. This was not how I had written the cow pens problem and I thought that they really should be related. I saw that the cow pen problem was 3 rows of walls minus the spaces that the cows occupy. So for the bull pens I looked for 3 rows of walls. All three rows were always of length equal to the number of bulls, plus one more than the number of bulls. This was because in the middle row of walls, you always have the one wall before each bull and the wall at the very end. This however created a solid figure leaving no room for the bulls, so you had to subtract out a number of walls equal to the number of bulls. Stage Number 1 2 3 4 5 100 n # of bulls 1 2 3 4 5 100 n Length of the Middle Section 1+2 2+3 3+4 4+5 5+6 100 + 101 n + (n +1) = 2n + 1 3 times the length 9 15 21 27 33 603 6n + 3 Number of walls 9–1=8 15 – 2 = 13 21 – 3 = 18 27 – 4 = 23 33 – 5 = 28 603 – 100 = 503 6n + 3 – n = 5n + 3