Assume you have 12 cows that live in a barn, which we call B. Call the set of th

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Assume you have 12 cows that live in a barn, which we call B. Call the set of those 12 cows C. We can define a sequence S = hcii that orders the cows from the firstborn to the youngest (i.e., c1 is the firstborn cow, and c12 is the youngest) — you can assume no two cows were born at the exact same time. Assume there is a field, which we call F, in which the cows graze. After the cows graze each day, they return to the barn B to sleep. You have an issue in that B gets very stinky when the cows return, particularly in summer. That said, z C, N(z), where the predicate N(x) means that x is not that impressed with you! After all, the barn B only has three stalls in which the cows can sleep. Assume that every cow must sleep in a stall, so that the cows do not get into trouble at night by trying to return to F.

(a) Prove that at least two of the cows were born on the same day of the week (where “day of the week” means one of Sunday, Monday, Tuesday, etc.).

(b) Prove that there is at least one stall that will be cramped tonight, where “cramped” is defined to mean that it is shared by at least three cows.

(c) You are going to buy 8 additional cows, but worry that then the cows will be very, very displeased with their sleeping conditions.

To try to prevent an organized revolt among your cows, you decide to find some more fields. Denote the current field, F, as F1. Then denote any additional fields that you are able to acquire as F2, F3, etc. Your plan is to separate all of the cows (both original and new taken together) into smaller groups during the day, and limit the occupancy of each field to no more than three cows. Your hope is that an organized insurrection among the cows will thus be prevented. Despite B being substandard, you do care for your cows. So, you will place a complete herd of three cows in as many of your fields as possible. Will each field have a full set of three cows, or will one field contain lonelier cows who do not have as many friends during the daytime? If there would have to be one field with fewer cows, how many would it have? Justify your answer using a single equation.

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