# As an example_ suppose you have a big bag of coins

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```					               Suppose you have a big
bag of coins

37 pennies
25 nickels
59 dimes              You will make a series
84 quarters
Value=\$65.15
of transactions using the
Number = 205           coins in your bag. Then
you will value the final
Exchange coins
according to value

37 pennies
??? pennies
25 nickles    There are a     ??? nickles
59 dimes      large number    ??? dimes
84 quarters   of possible     ??? quarters
Value=\$65.15
transactions:   Value=\$65.15
Number = 205                   Number = ???
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
Knowing these ‘value rules,’ can
we predict the final contents of the
bag?

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

37 pennies    As long as       57 pennies
25 nickels    everyone         33 nickels
59 dimes      played fairly,   58 dimes
84 quarters   the value of     82 quarters
Value=\$65.15   the bag          Value=\$65.15
Number = 205                    Number = ???
should not
change.
Suppose we exchange
coins one-for-one,
regardless of value

37 pennies                   ??? pennies
25 nickels    Exchange       ??? nickles
59 dimes      one coin for   ??? dimes
84 quarters   one coin,      ??? quarters
Value=\$65.15   regardless     Value = ???
Number = 205                  Number = 205
of value
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
Knowing the count of coins, can
we predict the contents of the
bag?

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

37 pennies                   40 pennies
25 nickels    Now it is      26 nickels
59 dimes      the number     59 dimes
84 quarters   of coins in    80 quarters
Value=\$65.15   the bag that   Value = ???
Number= 205                   Number= 205
has not
changed.
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.
If we do our coin transactions at
different times, do we expect the
same results?

Are the transaction rules
invariant with respect to
time?
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

```
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 views: 3 posted: 7/3/2011 language: English pages: 12