Day 13,
MAT115
Last Time
Next Time
Announcements:
Exam 1
is returned.
A key
is available.
There is a curve: add 7 points to your score. This is (partly) due to the complete misunderstanding of the Fraudini problem.
Question of the Day:
How do I win cash and other goodies by mastering Nim?
The Nim Game:
Rules:
There are two players, Player 1 and Player 2. Player 1 will go first.
An arbitrary number of counters is thrown down (e.g. toothpicks, pennies).
Player 1 must pick up at least one counter, but may not pick up all the counters.
Player 2 must then pick up at least one counter, and not more than twice the number of counters picked up on Player 1's turn.
The players continue taking turns, subject to these same conditions: pick up at least one, no more than twice what the preceding player took.
The player who picks up the final counter wins.
Let's play a sample game: Suppose that there are 27 counters on the table.
You play a sample game with a partner: Suppose that there are 43 counters on the table.
Now let's retreat to 13th century Italy for a moment: the bunnies of
Leonardo Pisano Fibonacci
Start with a new-born pair of "immortal" bunnies (they never die).
A new-born pair requires a month to mature.
The following month, the pair produces a new pair, which is subject to the same rules.
Do it again, do it again, do it again!
Fibonacci number decomposition and
Fibonacci Nim
.
If you looked at some special cases, what can we conclude?
Number of sticks
1
2
3
4
5
6
7
8
Winner
X
What's the winning strategy in Fibonacci Nim?
The Fibonacci numbers as a
recurrence relation
(do it again, do it again, do it again!).
More details on Fibonacci numbers and nim
. It turns out that
Every natural number is either
Fibonacci, or
can be written as a sum of
non-consecutive
Fibonacci's in a unique way.
Examples?
Fibonacci Numbers and Nature
:
Let's look at some pine cones....
dandelions (
Here's a scanned dandelion
)
Spirals in daisies:
21 spirals
34 spirals
together
Similar things happen in artichokes, pineapples, etc.
As we move toward the so-called "golden mean" and "golden rectangle", we'll start with a nice logarithmic spiral (see
this site
)
Links
:
A good reference on Mayan mathematics
Website maintained by
Andy Long
. Comments appreciated.