Fundamentos de Matemática · Aula 9 Fundamentos de Matemática 3. O cenário 0 Aula 9 Fundamentos...

Fundamentos de Matemática

Humberto José Bortolossi

Departamento de Matemática Aplicada

Universidade Federal Fluminense

Aula 9

27 de janeiro de 2014

Aula 9 Fundamentos de Matemática 1

O Problema dos Peixes Voadores


Problema formulado originalmente porJohn Conway.

Adaptação do contexto sugerida porCarlos Tomei.


O cenário

0


Colocando um peixe no nível 1

0

1



0

1



0

1

2



0

1

2

3



0

1

2

3

4



0

1

2

3

4

5


Colocando um peixe no nível 5: não é possível!

0

1

2

3

4

5


Como provar?

0

1

2

3

4

5


Definindo uma grade de energia

0

1

2

3

4

5x0



0

1

2

3

4

5x0 x1



0

1

2

3

4

5x0 x1 x2



0

1

2

3

4

5x0 x1 x2 x3



0

1

2

3

4

5x0x1 x1 x2 x3



0

1

2

3

4

5x0x1x2 x1 x2 x3



0

1

2

3

4

5x0x1x2x3 x1 x2 x3



0

1

2

3

4

5x0x1x2x3 x1 x2 x3

x4 x3 x1 x2x2 x4x3



0

1

2

3

4

5x0x1x2x3 x1 x2 x3

x4 x3 x1 x2x2 x4x3

x5 x4 x2 x3x3 x5x4



0

1

2

3

4

5x0x1x2x3 x1 x2 x3

x4 x3 x1 x2x2 x4x3

x5 x4 x2 x3x3 x5x4

x6 x5 x3 x4x4 x6x5



0

1

2

3

4

5x0x1x2x3 x1 x2 x3

x4 x3 x1 x2x2 x4x3

x5 x4 x2 x3x3 x5x4

x6 x5 x3 x4x4 x6x5

x7 x6 x4 x5x5 x7x6



0

1

2

3

4

5x0x1x2x3 x1 x2 x3

x4 x3 x1 x2x2 x4x3

x5 x4 x2 x3x3 x5x4

x6 x5 x3 x4x4 x6x5

x7 x6 x4 x5x5 x7x6

x8 x7 x5 x6x6 x8x7



0

1

2

3

4

5x0x1x2x3 x1 x2 x3

x4 x3 x1 x2x2 x4x3

x5 x4 x2 x3x3 x5x4

x6 x5 x3 x4x4 x6x5

x7 x6 x4 x5x5 x7x6

x8 x7 x5 x6x6 x8x7

x9 x8 x6 x7x7 x9x8



0

1

2

3

4

5x0x1x2x3 x1 x2 x3

x4 x3 x1 x2x2 x4x3

x5 x4 x2 x3x3 x5x4

x6 x5 x3 x4x4 x6x5

x7 x6 x4 x5x5 x7x6

x8 x7 x5 x6x6 x8x7

x9 x8 x6 x7x7 x9x8

x10 x9 x7 x8x8 x10x9



0

1

2

3

4

5x0x1x2x3 x1 x2 x3

x4 x3 x1 x2x2 x4x3

x5 x4 x2 x3x3 x5x4

x6 x5 x3 x4x4 x6x5

x7 x6 x4 x5x5 x7x6

x8 x7 x5 x6x6 x8x7

x9 x8 x6 x7x7 x9x8

x10 x9 x7 x8x8 x10x9

x11 x10 x8 x9x9 x11x10



0

1

2

3

4

5x0x1x2x3 x1 x2 x3

x4 x3 x1 x2x2 x4x3

x5 x4 x2 x3x3 x5x4

x6 x5 x3 x4x4 x6x5

x7 x6 x4 x5x5 x7x6

x8 x7 x5 x6x6 x8x7

x9 x8 x6 x7x7 x9x8

x10 x9 x7 x8x8 x10x9

x11 x10 x8 x9x9 x11x10

x12 x11 x9 x10x10 x12x11



0

1

2

3

4

5x0x1x2x3 x1 x2 x3

x4 x3 x1 x2x2 x4x3

x5 x4 x2 x3x3 x5x4

x6 x5 x3 x4x4 x6x5

x7 x6 x4 x5x5 x7x6

x8 x7 x5 x6x6 x8x7

x9 x8 x6 x7x7 x9x8

x10 x9 x7 x8x8 x10x9

x11 x10 x8 x9x9 x11x10

x12 x11 x9 x10x10 x12x11

x13 x12 x10 x11x11 x13x12



0

1

2

3

4

5x0x1x2x3 x1 x2 x3

x4 x3 x1 x2x2 x4x3

x5 x4 x2 x3x3 x5x4

x6 x5 x3 x4x4 x6x5

x7 x6 x4 x5x5 x7x6

x8 x7 x5 x6x6 x8x7

x9 x8 x6 x7x7 x9x8

x10 x9 x7 x8x8 x10x9

x11 x10 x8 x9x9 x11x10

x12 x11 x9 x10x10 x12x11

x13 x12 x10 x11x11 x13x12

x14 x13 x11 x12x12 x14x13


A energia deste peixe é x5

0

1

2

3

4

5x0x1x2x3 x1 x2 x3

x4 x3 x1 x2x2 x4x3

x5 x4 x2 x3x3 x5x4

x6 x5 x3 x4x4 x6x5

x7 x6 x4 x5x5 x7x6

x8 x7 x5 x6x6 x8x7

x9 x8 x6 x7x7 x9x8

x10 x9 x7 x8x8 x10x9

x11 x10 x8 x9x9 x11x10

x12 x11 x9 x10x10 x12x11

x13 x12 x10 x11x11 x13x12

x14 x13 x11 x12x12 x14x13


A energia deste cardume é x5 + x6 + x7

0

1

2

3

4

5x0x1x2x3 x1 x2 x3

x4 x3 x1 x2x2 x4x3

x5 x4 x2 x3x3 x5x4

x6 x5 x3 x4x4 x6x5

x7 x6 x4 x5x5 x7x6

x8 x7 x5 x6x6 x8x7

x9 x8 x6 x7x7 x9x8

x10 x9 x7 x8x8 x10x9

x11 x10 x8 x9x9 x11x10

x12 x11 x9 x10x10 x12x11

x13 x12 x10 x11x11 x13x12

x14 x13 x11 x12x12 x14x13


Como devemos escolher o valor de x?

Antes Depois Diferença de Energia

xn+2 xn+1 xn

xn − xn+1 − xn+2

xn+1 xn xn+1

−xn

xn xn+1 xn+2

xn+2 − xn+1 − xn

1− x − x2 = 0 ⇒ x =−1−

√5

2ou x =

−1 +√

52

.




xn+2 xn+1

xn

xn − xn+1 − xn+2

xn+1 xn xn+1

−xn

xn xn+1 xn+2

xn+2 − xn+1 − xn

1− x − x2 = 0 ⇒ x =−1−

√5

2ou x =

−1 +√

52

.




xn+2 xn+1 xn

xn − xn+1 − xn+2

xn+1 xn xn+1

−xn

xn xn+1 xn+2

xn+2 − xn+1 − xn

1− x − x2 = 0 ⇒ x =−1−

√5

2ou x =

−1 +√

52

.




xn+2 xn+1 xn

xn − xn+1 − xn+2

xn+1 xn

xn+1

−xn

xn xn+1 xn+2

xn+2 − xn+1 − xn

1− x − x2 = 0 ⇒ x =−1−

√5

2ou x =

−1 +√

52

.




xn+2 xn+1 xn

xn − xn+1 − xn+2

xn+1 xn xn+1

−xn

xn xn+1 xn+2

xn+2 − xn+1 − xn

1− x − x2 = 0 ⇒ x =−1−

√5

2ou x =

−1 +√

52

.




xn+2 xn+1 xn

xn − xn+1 − xn+2

xn+1 xn xn+1

−xn

xn xn+1

xn+2

xn+2 − xn+1 − xn

1− x − x2 = 0 ⇒ x =−1−

√5

2ou x =

−1 +√

52

.




xn+2 xn+1 xn

xn − xn+1 − xn+2

xn+1 xn xn+1

−xn

xn xn+1 xn+2

xn+2 − xn+1 − xn

1− x − x2 = 0 ⇒ x =−1−

√5

2ou x =

−1 +√

52

.




xn+2 xn+1 xn

xn(1− x − x2)

xn+1 xn xn+1

−xn

xn xn+1 xn+2

xn(x2 − x − 1)

1− x − x2 = 0 ⇒ x =−1−

√5

2ou x =

−1 +√

52

.




xn+2 xn+1 xn

xn(1− x − x2)

xn+1 xn xn+1

−xn

xn xn+1 xn+2

xn(x2 − x − 1)

x =−1 +

√5

2é > 0 e < 1.




xn+2 xn+1 xn

xn(1− x − x2)

xn+1 xn xn+1

−xn

xn xn+1 xn+2

xn(x2 − x − 1)

xn(1− x − x2) = 0




xn+2 xn+1 xn

xn(1− x − x2)

xn+1 xn xn+1

−xn

xn xn+1 xn+2

xn(x2 − x − 1)

−xn < 0




xn+2 xn+1 xn

xn(1− x − x2)

xn+1 xn xn+1

−xn

xn xn+1 xn+2

xn(x2 − x − 1)

xn(x2 − x − 1)= xn(1− x − x − 1)= −2 xn+1 < 0




xn+2 xn+1 xn

xn(1− x − x2)

xn+1 xn xn+1

−xn

xn xn+1 xn+2

xn(x2 − x − 1)

Moral: em cada jogada, a energia de um cardume nunca aumenta!


Qual é a energia de todo o oceano?

0

1

2

3

4

5

x8 x7 x5 x6x6 x8x7

x9 x8 x6 x7x7 x9x8

x10 x9 x7 x8x8 x10x9

x11 x10 x8 x9x9 x11x10

x12 x11 x9 x10x10 x12x11

x13 x12 x10 x11x11 x13x12

x14 x13 x11 x12x12 x14x13



x5 + x6 + x7 + · · ·

+

2 ·(

x6 + x7 + x8 + · · ·)

+

2 ·(

x7 + x8 + x9 + · · ·)

+

...



x5 ·(

1 + x + x2 + · · ·)

+

2 x6 ·(

1 + x + x2 + · · ·)

+

2 x7 ·(

1 + x + x2 + · · ·)

+

...



x5 · 11− x+

2 x6 · 11− x+

2 x7 · 11− x+

...



x5 · 11− x

+ 2 x6 · 11− x

+ 2 x7 · 11− x

+ · · ·

=

x5

1− x+

2 x6

1− x· (1 + x + x2 + · · · )

=

x5

1− x+

2 x6

1− x· 1

1− x

=x5

x2 +2 x6

x2 ·1x2

=

x3 + 2 x2



x3 + 2 x2

=

x · x2 + 2 x2

=

x(1− x) + 2 x2

=

x − x2 + 2 x2

=x + x2

=

x + 1− x

=

1


Não é possível colocar um peixe no nível 5!

0

1

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5x0


Referência:

http://www.cut-the-knot.org/proofs/checker.shtml


Seção de Exercícios


Fundamentos de Matemática · Aula 9 Fundamentos de Matemática 3. O cenário 0 Aula 9 Fundamentos...

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