Sample Problems

1. Sample Problems • Ted’s weekly income is $100.00 less than double Gary’s weekly income. If Ted makes $500.00 a week, what does Gary make? • Paul makes $25.00 a week less than the sum of what Fred and Carl together make. Carl’s weekly income would be triple Steven’s if he made $50.00 more a week. Paul makes $285.00 a week and Steven makes $75.00 a week. How much does Fred make?

2. Investigation and Argument • Solving a problem has two phases: • Investigation: Find a solution • Argument: Get the solution across to a “client” • Too often, we only see the polished argument in a book • Hopefully this course helps you with investigation • But you also have to be good at “argument”

3. Argument • Bad argument: • Coercing through force of will or personality • Irrespective of correctness • Good argument: • Clear presentation • Logical progression of steps • Enough and not too much • Successful

4. When Do You “Argue” • Anytime you write • Email • Letters • Tests • Homework • Proposals • Mechanisms: • Proofs • Essays, papers, books

5. Chessboard Problem Problem: Completely tile defective chessboard with dominos

6. Chessboard Problem Claim: Tiling the defective chessboard with dominos is impossible. Proof? Must be a convincing argument

7. First Proof Attempt There are more black squares than white squares. Therefore, tiling the defective chessboard with dominos is impossible. Why is this not an adequate argument?

8. Second Proof Attempt Every domino covers one black square and one white square. The defective chessboard has two more black squares than white squares. Therefore, any tiling with dominos will leave at least two black squares uncovered. We conclude that tiling the defective chessboard with dominos is impossible. Is this an adequate argument?