Fibonacci Sequence, Golden Ratio and Wythoff's

GameLOEKY HARYANTO

Mathematics Department, Hasanuddin University, email: L.Haryanto@unhas.ac.id

HaryantoL@outlook.comGSM’s #: +6281342127598

A Relationship among Fibonacci Sequence, Fibonacci Word, P-positions of Wythoff’s game and the Golden Ratio

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USE THIS PRESENTATION AS A NEW STRATEGY FOR STUDENT-CENTERED LEARNING (SCL) METHOD.

The Maplet copies here were created to make students firstly being familiar with (not necessarily mastering the theory of) combinatorial game and its related topics before the students being introduced with the theoretical parts of the subject; e.g. before they were given some formal theories which were written in the next page!

By the way, since mathematics is a language which is full of written symbols, without visual and ‘seemingly’ interactive presentations, most of students tend to sleep in abstract algebra classes. Nevertheless, IMO most strategies proposed for the SCL method by experts in education are not appropriate for math classes, or even worse than the common usual (old) teaching method.

Relationships: Fibonacci word (automata theory):

Alphabet: A = {a, b}; morphism on monoid A*

(a) = ab, (b) = a and (u,v A*)[(uv) = (u)(v)] Fibonacci Sequence (discrete math, combinatorics)

F0 = 1, F1 = 2 and (n > 2)[Fn = Fn 1 + Fn 2] or

using the Binet’s formula Fn =

where = is the golden ratio Complementary Beatty Sequence (recreational math)

{[n(1 + 1/])}n1 {([n(1 + )]}n1 =

&

{[n(1 + 1/])}n1 {([n(1 + )] }n1 =

More relationships (graph theory, fractal theory, dynamical theory, etc)

( )5

n n

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The next Maplet simulates 12 iterations of the morphism : 0(a), 1(a), …, 12(a). It shows:1. the word produces at each morphic iteration;2. the sequence of word lengths produced by the

done morphic iterations;3. the P-positions of the Wythoff’s game.

Complementary Beatty’s Sequences Particular: P-Positions of Wythoff’s game

{[n(1 + 1/])}n1 {([n(1 + )]}n1 =

&

{[n(1 + 1/])}n1 {([n(1 + )] }n1 =

where = 1.6180339887… is the golden ratio General: Given irrational x (0, 1)

{([n(1 + x)]}n1 {[n(1 + 1/x])}n1 =

&

{([n(1 + x)] }n1 {[n(1 + 1/x])}n1=

Advantages of Maplet’s observations, e.g. for any irrational x with 0 < x < 1,

1 1lim (1 ) 2 1 & lim (1 1/ ) 2x x

n x n n x n

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Next Maplet Output gives 1. The first twenty-one P-positions (ai, bi) of Wythoff’s game where

ai = floor(i*(1 + 1/rho)), bi = floor(i*(1 + rho)),

2. The first twenty-one elements of Beatty’s sequence (Ai, Bi) where

Ai = floor(i*(1 + 1/x)), Bi = floor(i*(1 + x)),

for some x = 1 + y > 1.

Theorem

Let x > 0 be an irrational number and y = 1/x, then the two sequences positive integers

{floor(i(1 + x))}i 1 and {floor(i(1 + y))}i 1 are two complementary sequences;

i.e.

1. {floor(i(1 + x))}i 1 {floor(i(1 + y))}i 1 = .

2. {floor(i(1 + x))}i 1 {floor(i(1 + y))}i 1 = N.

Nothing extra ordinary on the second pair of sequences. The Maplet outputs only illustrate that if {xi} is a sequence of positive irrational numbers sastifying lim xi = 0 then the previous observation works. It is equivalent with

So those

11lim (1 ) 2 1 & lim (1 (1 )) 2

i ii ixn n n x n

The irrational properties of xi cannot be dropped as the following Maplet examples show that in case xis are rationals, the two resulting sequences may not be disjoint. In the first example, they share the integer 53 (and many other integers) whereas in the second example, they share the integer 116.

The Wythoff’s Game P-Positions of Wythoff’s game are the pairs ([[n(1 + 1/], n(1 + )]).

Theorem:

{[n(1 + 1/])}n1 {([n(1 + )]}n1 =

&

{[n(1 + 1/])}n1 {([n(1 + )] }n1 =

where = 0.6180339887… is the golden ratio.

Wythoff’s game is played by two players that alternatingly remove some tokens from two heaps according to the following rules:

1. Each player can remove any positive number of tokens from one of the two heaps.

2. Each player can remove the same number of tokens from both heaps.

3. A player is lost the game if there is no token left for, or to be removed by the player.

Equivalently, a player who takes the last tokens is the winner.

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Notice that once the computer has made automatic moves, you were forced to respond the following pairs of tokens’ numbers:

1. (74, 120);2. (27, 44);3. (4, 7);4. (1, 2);5. (0, 0).

These pairs belong to the P-positions of Wythoff’s game. The pair (27, 44) can be seen at the Maplets’ 7-8th morphic iterations whereas the pair (74, 120) can be seen at the Maplet’s 9th morphic iteration.