1. 37 pennies 25 nickels 59 dimes 84 quarters Value=$65.15 Number = 205 You will make a series of transactions using the coins in your bag. Then you will value the final contents of your bag. Suppose you have a big bag of coins

2. 37 pennies 25 nickles 59 dimes 84 quarters Value=$65.15 Number = 205 ??? pennies ??? nickles ??? dimes ??? quarters Value=$65.15 Number = ??? Exchange coins according to value There are a large number of possible transactions:

3. Suppose you make this series of transactions 1.Replace 1 nickel by 5 pennies 2.Replace 1 quarter by 3 nickels and 1 dime 3.Replace 1 dime by 1 nickel and 5 pennies 4.Replace 1 quarter by 25 pennies 5.Replace 1 quarter by 5 nickels 6.Replace 1 dime and 15 pennies by 1 quarter

4. 37 pennies 25 nickels 59 dimes 84 quarters Value=$65.15 Number = 205 ??? pennies ??? nickels ??? dimes ??? quarters Value=$65.15 Number = ??? As long as everyone played fairly, the value of the bag should not change. Knowing these ‘value rules,’ can we predict the final contents of the bag?

5. 37 pennies 25 nickels 59 dimes 84 quarters Value=$65.15 Number = 205 57 pennies 33 nickels 58 dimes 82 quarters Value=$65.15 Number = ??? As long as everyone played fairly, the value of the bag should not change. Knowing these ‘value rules,’ can we predict the final contents of the bag?

6. 37 pennies 25 nickels 59 dimes 84 quarters Value=$65.15 Number = 205 ??? pennies ??? nickles ??? dimes ??? quarters Value = ??? Number = 205 Suppose we exchange coins one-for-one, regardless of value Exchange one coin for one coin, regardless of value

7. Make this series of transactions 1.Replace 1 nickel by 1 penny 2.Replace 1 quarter by 1 nickel 3.Replace 1 dime by 1 penny 4.Replace 1 quarter by 1 penny 5.Replace 1 quarter by 1 nickel 6.Replace 1 quarter by 1 dime

8. 37 pennies 25 nickels 59 dimes 84 quarters Value=$65.15 Number= 205 ??? pennies ??? nickels ??? dimes ??? quarters Value = ??? Number= 205 Now it is the number of coins in the bag that has not changed. Knowing the count of coins, can we predict the contents of the bag?

9. 37 pennies 25 nickels 59 dimes 84 quarters Value=$65.15 Number= 205 40 pennies 26 nickels 59 dimes 80 quarters Value = ??? Number= 205 Now it is the number of coins in the bag that has not changed. Knowing the count of coins, can we predict the contents of the bag?

10. Suppose we do our coin transactions in different places (but we always follow the same rules). Do we expect the same results each time? In other words: Are the transaction rules invariant with respect to position in space? Our coin experiment is self- contained, so it should not vary with position.

11. If we do our coin transactions at different times, do we expect the same results? Are the transaction rules invariant with respect to time?

12. If we do our coin transactions in different orders, do we expect the same results? Do the transactions rules produce constant results with respect to order (who goes first)? Back