- Your exam results show how important homework is:
- Test score = 0.714874*(Homework average from 100) + 16.5795
- Test score = 0.714874*(Homework average from 100) + 16.5795
- You have an assignment to hand in, due next Thursday.
- You have a reading assignment for next time. Please read Golden
Rectangles (Idea #12, p. 48).
- Your bases homework is returned; sorry, I'm still working on your Mayan, Babylonian, Lewis Carrollinian homework....
- 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.
- We looked at some special cases and what did we
conclude?
Number of coins 1 2 3 4 5 6 7 8 Winner X P2 P2 P1 P2 P1 P1 P2 - What's the winning strategy in Fibonacci Nim?
- If the number of coins is Fibonacci, go second;
- If the number of coins is not Fibonacci, go first.
- At your turn, write the number as the unique sum
of non-consecutive Fibonacci numbers, then take the
smallest Fibonacci in the sum.
If you can apply this strategy, you are destined to win -- if you can add Fibonacci numbers successfully....
- Every natural number is either
- Fibonacci, or
- can be written as a sum of non-consecutive Fibonacci's in a unique way.
- 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!
- Here's a tree showing this.
- Let's look at some pine cones....
- But this is far more general than pine cones. Let's have a word from Vi Hart. (She was the one who does the binary music.)
- Dandelions (Here's a random naked dandelion that I scanned.) Notice that each pore is on a spiral arm. There's no skipping pores!
- Spirals in daisies:
