Deduction

MATHEMATICAL PROOF

COINS & CLIPS

                       


                       

1. Look at the squares on which the coins were placed on each board.
   How do their colors seem to be related to who wins?
   
   
2. If you conclude from these eight examples
   that your answer to the first question is correct for every possible game,
   what kind of reasoning are you using?
   
   
3. Do the eight examples prove it?
   
   
4. According to the rules,
   each paperclip has to be placed on two squares that share a common side.
   What can be concluded about the colors of any two such squares?
   
   
5. If player B is able to place seven paperclips so that they cover fourteen squares,
   what must be true about the colors of those fourteen squares?
   
   Of the fourteen remaining squares,
   how many squares are there of each color if player A begins by placing the coins on
   
   
6. two yellow squares?
   
   
7. two red squares?
   
   
8. a yellow square and a red square?
   
   
9. What kind of reasoning are you using when you conclude from your previous answers
   that your answer to exercise 1 is correct for every possible game?
   
   

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   MARBLES & MATCHBOXES

   
   Many puzzles, including the following one about marbles in matchboxes,
   can be solved by reasoning deductively.
   The figure below shows three matchboxes.
   One contains two red marbles,
   one contains two white marbles,
   and one contains a red marble and a white marble.
   The labels telling the contents of the boxes have been switched, however,
   so that the label on each box is wrong.
   You are permitted to choose one box and open it far enough to see just one marble.
   The challenge is to explain how you can figure out from this what is in each box.
   
   
   
   
   
   Suppose that you choose the box labeled “2 red.”
   
   
10. If you opened it and saw a red marble,
    what color would the other marble have to be and how do you know?
    
    
11. If you opened it and saw a white marble, could the other marble be white? Could it be red?
    
    
    Suppose instead that you choose the box labeled “2 white.”
    
12. If you opened it and saw a white marble, what could you conclude?
    
    
13. If you opened it and saw a red marble, would you know the color of the other marble?
    
    
    Suppose instead that you choose the box labeled “1 red, 1 white.”
    
    
14. If you opened it and saw a red marble, what could you conclude?
    
    
15. If you opened it and saw a white marble, what could you conclude?
    
    On the basis of your answers so far, one box would be the best choice to look inside.
    
    
16. Which one?
    
    
    Suppose that you see a red marble when you open the box labeled “1 red, 1 white.”
    What can you conclude about
    
    
17. the other marble in the box?
    
    
18. the marbles in the box labeled “2 white”?
    
    
19. the marbles in the box labeled “2 red”?
    
    

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    MINIMALL

    
    
    The local minimall includes an auto repair, a burger drive-thru, a CD swapshop, and a drug store. All the stores are built around a quadrangle with a courtyard in the middle.
    
    
    
    1. Al owns the burger drive-thru.
    2. Al's burger drive-thru is to the left of Chuck's store.
    3. Bill's store is to the right of the drug store.
    4. Dave's store faces Chuck's store.
    5. Dave's store is not the auto shop.
    
    
20. This puzzle consists of five facts. Assume they are all true. Use these facts to deduce what kind of store each person has. Complete the table above as you would your detective notes in the game of Clue.
    
    
21. Then fill in the color squares below showing where each store and its owner is located in the minimall.
    
    
    
    
    

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    NUMBER TRICKS

    
    
    The following number trick is illustrated with two different numbers.
    Proofs that it will always work are shown with $'s and #'s,
    and with algebraic symbols.
    
    
    

    	EXAMPLE 1	EXAMPLE 2	PROOF 1 with $ and #	PROOF2 with n
    Choose a number	5	8	$	n
    Multiply by 3	5x3=15	8x3=24	$$$	3n
    Add 6	15+6=21	24+6=30	$$$######	3n+6
    Divide by 3	21/3=7	30/3=10	$##	n+2
    Subtract the number first thought of	7-5=2	10-8=2	##	n+2-n
    The result is 2				2

    
    Complete the number tricks in the tables below
    using the same format as in the example columns above.
    Then write two proofs that show why the trick will always work:
    one proof with $'s and #'s, and one with n and algebraic symbols.
    The chosen number is the number first thought of on row 1.
    
    
22.  
    
    
    
    
23.  
    

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    The following number trick is of a different type from those above.
    
    Choose any three-digit number.
    Multiply it by 7.
    Multiply the result by 11.
    Multiply the result by 13.
    
    An example of carrying out these steps is shown below.
    
    524
    524 x 7= 3,668
    3,668 x 11 = 40,348
    40,348 x 13 = 524,524
    
    
24. Do the trick again with a different three-digit number. Show all four steps.
    
    
25. What relation does the final result seem to have to the number that was originally chosen?
    
    
26. Suppose that you did this trick with many different three digit numbers and it always worked.
    What kind of reasoning would you be using if you concluded that it will work for every three-digit number?
    
    
27. Find 7 x 11 x 13.
    
    The number 1,001 is the key to this number trick.
    
    
28. Without using a calculator,
    multiply the three-digit number that you chose for exercise 1 by 1,001.
    Show your work.
    
    
29. Multiplying a number by 7, 11, and 13 in succession
    gives the same result as multiplying it by what number?
    
    
30. Does this seem to suggest that the trick will work for every three-digit number?