Day 18, MAT115

They generate the numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ....

• Rules: how to play Fibonacci Nim
  • 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.

• Opening position in Fibonacci Nim

  When we looked at some special cases, what did we conclude?

  Number of sticks	1	2	3	4	5	6	7	8
  Winner	X	P2	P2	P1	P2	P1	P1	P2

  It appears that you want to be Player 2 if the starting number of counters is Fibonacci, and Player 1 otherwise.

• The secret to the play of Fibonacci nim.

  It turns out that every natural number is either

  1. Fibonacci, or
  2. can be written as a sum of non-consecutive Fibonacci numbers in a unique way.

  So, for example, 28=21+5+2.

  More examples? Write the following as sums of non-consecutive Fibonacci numbers:

  • 19
  • 73
  • 131

  The secret to the play is this: if you're looking at a non-Fibonacci number, write it as a sum of non-consecutive Fibonacci numbers, then take the smallest Fibonacci number of counters in the sum.

  So, for example, if the sum is 28=21+5+2 to you, then since you are allowed to take 2 counters, you're guaranteed to win (if you keep using the strategy).

  Notice that the other player would be looking at 26=21+5, and wouldn't be allowed to use the strategy -- they can't take five!

• What's the winning strategy in Fibonacci Nim?
  • First of all start in the right position (player 1 or player 2), if you can. Then choose the smallest Fibonacci from the sum of non-consecutive Fibonaccis.
  • If you're in the wrong position, stall by taking just 1 counter; if someone doesn't know what they're doing, they'll eventually screw up and put you in the driver's seat.

• Let's practice playing some sample games:
  • Start with 15 counters; try being both player 1 and player 2 with your partner.
  • Try starting with 42 counters. Which player do you want to be? What's your first move?

As we move toward the concepts of the so-called "golden mean" and "golden rectangle", we'll start with a nice Fibonacci spiral (see this site).

The process of creating a Fibonacci spiral is simple:

• Start: Draw a square.
• To each rectangle, attach the largest square possible to make a new rectangle. Make sure that you always proceed in the same direction (clockwise, or counter-clockwise).
• Do it again, do it again, do it again....

Make sure that you send me the final image, and not just a link (that won't do anything!). Right click on the image, and save it to your computer -- then past the image into your email. That's the easiest way, I think.