Proof by CrayonVirginia Bohme

Ji LiThe University of Arizona

Tucson Middle School Teachers’ CircleMEAD Jan 30, 2010

Proof by Crayon Tucson MS Teacher’s Circle

Tile this 4 x 4 grid with 1 x 2 dominoes

Tile this board with 1 x 2 dominoes

Proof by Crayon Tucson MS Teacher’s Circle

Tile this board with 1 x 2 dominoes

Proof by Crayon Tucson MS Teacher’s Circle

How many dominoes are required?

Tile this board with 1 x 2 dominoes

Proof by Crayon Tucson MS Teacher’s Circle

Tile this board with 1 x 2 dominoes

Proof by Crayon Tucson MS Teacher’s Circle

Try some other boards with two squares removed.

Tile this board with 1 x 2 dominoes

Proof by Crayon Tucson MS Teacher’s Circle

Proof by Crayon Tucson MS Teacher’s Circle

Proof by Crayon Tucson MS Teacher’s Circle

Proof by Crayon Tucson MS Teacher’s Circle

Proof by Crayon Tucson MS Teacher’s Circle

Proof by Crayon Tucson MS Teacher’s Circle

Proof by Crayon Tucson MS Teacher’s Circle

Proof by Crayon Tucson MS Teacher’s Circle

Proof by Crayon Tucson MS Teacher’s Circle

Proof by Crayon Tucson MS Teacher’s Circle

Proof by Crayon Tucson MS Teacher’s Circle

Impossible!

Proof by Crayon Tucson MS Teacher’s Circle

This grid is also impossible. But the method of “forcing” is horrible to consider...

Proof by Crayon Tucson MS Teacher’s Circle

This grid is also impossible. But the method of “forcing” is horrible to consider...

New ideas?

Proof by Crayon Tucson MS Teacher’s Circle

Proof by Crayon Tucson MS Teacher’s Circle

Proof by Crayon Tucson MS Teacher’s Circle

Proof by Crayon Tucson MS Teacher’s Circle

Proof by Crayon Tucson MS Teacher’s Circle

Proof by Crayon Tucson MS Teacher’s Circle

There are 8

Proof by Crayon Tucson MS Teacher’s Circle

There are 8

Proof by Crayon Tucson MS Teacher’s Circle

There are 8

There are 6

Proof by Crayon Tucson MS Teacher’s Circle

There are 8

There are 6

Proof by Crayon Tucson MS Teacher’s Circle

There are 8

There are 6

Impossible!

Proof by Crayon Tucson MS Teacher’s Circle

There are 8

There are 6

Impossible!

0 1 0 1

1 0 1 0

0 1 0 1

1 0 1 0

Proof by Crayon Tucson MS Teacher’s Circle

This grid is also impossible.

Does the new method of “coloring” work?

Proof by Crayon Tucson MS Teacher’s Circle

This grid is also impossible.

Does the new method of “coloring” work?

0 1 0 1 0 1 0 11 0 1 0 1 0 1 00 1 0 1 0 1 0 11 0 1 0 1 0 1 00 1 0 1 0 1 0 11 0 1 0 1 0 1 00 1 0 1 0 1 0 11 0 1 0 1 0 1 0

Proof by Crayon Tucson MS Teacher’s Circle

In how many different ways can you tile this 2x8 board with dominoes?

Proof by Crayon Tucson MS Teacher’s Circle

In how many different ways can you tile this 2x8 board with dominoes?

Proof by Crayon Tucson MS Teacher’s Circle

In how many different ways can you tile this 2x8 board with dominoes?

Proof by Crayon Tucson MS Teacher’s Circle

Proof by Crayon Tucson MS Teacher’s Circle

In how many different ways can you tile a smaller board?

Proof by Crayon Tucson MS Teacher’s Circle

Proof by Crayon Tucson MS Teacher’s Circle

Proof by Crayon Tucson MS Teacher’s Circle

Proof by Crayon Tucson MS Teacher’s Circle

Proof by Crayon Tucson MS Teacher’s Circle

Proof by Crayon Tucson MS Teacher’s Circle

Proof by Crayon Tucson MS Teacher’s Circle

Proof by Crayon Tucson MS Teacher’s Circle

Proof by Crayon Tucson MS Teacher’s Circle

Proof by Crayon Tucson MS Teacher’s Circle

Proof by Crayon Tucson MS Teacher’s Circle

Proof by Crayon Tucson MS Teacher’s Circle

Proof by Crayon Tucson MS Teacher’s Circle

Proof by Crayon Tucson MS Teacher’s Circle

Proof by Crayon Tucson MS Teacher’s Circle

Proof by Crayon Tucson MS Teacher’s Circle

Proof by Crayon Tucson MS Teacher’s Circle

Proof by Crayon Tucson MS Teacher’s Circle

Proof by Crayon Tucson MS Teacher’s Circle

Proof by Crayon Tucson MS Teacher’s Circle

Proof by Crayon Tucson MS Teacher’s Circle

Proof by Crayon Tucson MS Teacher’s Circle

Proof by Crayon Tucson MS Teacher’s Circle

Proof by Crayon Tucson MS Teacher’s Circle

Proof by Crayon Tucson MS Teacher’s Circle

Proof by Crayon Tucson MS Teacher’s Circle

Proof by Crayon Tucson MS Teacher’s Circle

5=2+3

Proof by Crayon Tucson MS Teacher’s Circle

In how many different ways can you tile this board with dominoes?

Proof by Crayon Tucson MS Teacher’s Circle

In how many different ways can you tile this board with dominoes?

Last piece is vertical

Proof by Crayon Tucson MS Teacher’s Circle

In how many different ways can you tile this board with dominoes?

Last piece is vertical

Proof by Crayon Tucson MS Teacher’s Circle

In how many different ways can you tile this board with dominoes?

Last piece is vertical

Proof by Crayon Tucson MS Teacher’s Circle

In how many different ways can you tile this board with dominoes?

Last piece is vertical

Last piece is horizontal

Proof by Crayon Tucson MS Teacher’s Circle

In how many different ways can you tile this board with dominoes?

Last piece is vertical

Last piece is horizontal

Proof by Crayon Tucson MS Teacher’s Circle

In how many different ways can you tile this board with dominoes?

Last piece is vertical

Last piece is horizontal

Proof by Crayon Tucson MS Teacher’s Circle

In how many different ways can you tile a 2 x n board with dominoes?

... ... ...

... ... ...

Proof by Crayon Tucson MS Teacher’s Circle

In how many different ways can you tile a 2 x n board with dominoes?

... ... ...

... ... ...

Call this number f(n)

Proof by Crayon Tucson MS Teacher’s Circle

In how many different ways can you tile a 2 x n board with dominoes?

... ... ...

... ... ...

n

f(n)

Call this number f(n)

Proof by Crayon Tucson MS Teacher’s Circle

In how many different ways can you tile a 2 x n board with dominoes?

... ... ...

... ... ...

n

f(n)

Call this number f(n)

n 2 3 4 5 6 7 8

f(n)

Proof by Crayon Tucson MS Teacher’s Circle

In how many different ways can you tile a 2 x n board with dominoes?

... ... ...

... ... ...

n

f(n)

Call this number f(n)

n 2 3 4 5 6 7 8

f(n)

n 2 3 4 5 6 7 8

f(n) 2 3

Proof by Crayon Tucson MS Teacher’s Circle

In how many different ways can you tile a 2 x n board with dominoes?

... ... ...

... ... ...

n

f(n)

Call this number f(n)

n 2 3 4 5 6 7 8

f(n)

n 2 3 4 5 6 7 8

f(n) 2 3

n 2 3 4 5 6 7 8

f(n) 2 3 5

Proof by Crayon Tucson MS Teacher’s Circle

In how many different ways can you tile a 2 x n board with dominoes?

... ... ...

... ... ...

n

f(n)

Call this number f(n)

n 2 3 4 5 6 7 8

f(n)

n 2 3 4 5 6 7 8

f(n) 2 3

n 2 3 4 5 6 7 8

f(n) 2 3 5

n 2 3 4 5 6 7 8

f(n) 2 3 5 8

Proof by Crayon Tucson MS Teacher’s Circle

In how many different ways can you tile a 2 x n board with dominoes?

... ... ...

... ... ...

n

f(n)

Call this number f(n)

n 2 3 4 5 6 7 8

f(n)

n 2 3 4 5 6 7 8

f(n) 2 3

n 2 3 4 5 6 7 8

f(n) 2 3 5

n 2 3 4 5 6 7 8

f(n) 2 3 5 8

n 2 3 4 5 6 7 8

f(n) 2 3 5 8 13

Proof by Crayon Tucson MS Teacher’s Circle

In how many different ways can you tile a 2 x n board with dominoes?

... ... ...

... ... ...

n

f(n)

Call this number f(n)

n 2 3 4 5 6 7 8

f(n)

n 2 3 4 5 6 7 8

f(n) 2 3

n 2 3 4 5 6 7 8

f(n) 2 3 5

n 2 3 4 5 6 7 8

f(n) 2 3 5 8

n 2 3 4 5 6 7 8

f(n) 2 3 5 8 13

n 2 3 4 5 6 7 8

f(n) 2 3 5 8 13 21

Proof by Crayon Tucson MS Teacher’s Circle

In how many different ways can you tile a 2 x n board with dominoes?

... ... ...

... ... ...

n

f(n)

Call this number f(n)

n 2 3 4 5 6 7 8

f(n)

n 2 3 4 5 6 7 8

f(n) 2 3

n 2 3 4 5 6 7 8

f(n) 2 3 5

n 2 3 4 5 6 7 8

f(n) 2 3 5 8

n 2 3 4 5 6 7 8

f(n) 2 3 5 8 13

n 2 3 4 5 6 7 8

f(n) 2 3 5 8 13 21

n 2 3 4 5 6 7 8

f(n) 2 3 5 8 13 21 34

Proof by Crayon Tucson MS Teacher’s Circle

In how many different ways can you tile this 1xn board with and ?

Proof by Crayon Tucson MS Teacher’s Circle

In how many different ways can you tile this 1xn board with and ?

Proof by Crayon Tucson MS Teacher’s Circle

Again, we consider smaller boards to tile.

In how many different ways can you tile this 1xn board with and ?

Proof by Crayon Tucson MS Teacher’s Circle

Again, we consider smaller boards to tile.

In how many different ways can you tile this 1xn board with and ?

Proof by Crayon Tucson MS Teacher’s Circle

Again, we consider smaller boards to tile.

In how many different ways can you tile this 1xn board with and ?

Proof by Crayon Tucson MS Teacher’s Circle

Again, we consider smaller boards to tile.

In how many different ways can you tile this 1xn board with and ?

Proof by Crayon Tucson MS Teacher’s Circle

In how many different ways can you tile this 1xn board with and ?

Proof by Crayon Tucson MS Teacher’s Circle

Let this number be f(n). These numbers follow the same pattern as before..

In how many different ways can you tile this 1xn board with and ?

Proof by Crayon Tucson MS Teacher’s Circle

Let this number be f(n). These numbers follow the same pattern as before..

n 2 3 4 5 6 7 8

f(n) 2 3 5 8 13 21 34

Proof by Crayon Tucson MS Teacher’s Circle

Define a 12-composition of a number n is a way of writing n as the sum of 1’s and 2’s. For example,

3=1+2 is a 12-composition of 3.

Proof by Crayon Tucson MS Teacher’s Circle

Define a 12-composition of a number n is a way of writing n as the sum of 1’s and 2’s. For example,

3=1+2 is a 12-composition of 3.

How many 12-compositions are there for n?

Proof by Crayon Tucson MS Teacher’s Circle

How many 12-compositions are there for n?

n

1

countcompositions of n

234

Proof by Crayon Tucson MS Teacher’s Circle

How many 12-compositions are there for n?

n

1

countcompositions of n

1=1234

Proof by Crayon Tucson MS Teacher’s Circle

How many 12-compositions are there for n?

n

1

countcompositions of n

1=1 1234

Proof by Crayon Tucson MS Teacher’s Circle

How many 12-compositions are there for n?

n

1

countcompositions of n

1=1 1234

2=1+1=2

Proof by Crayon Tucson MS Teacher’s Circle

How many 12-compositions are there for n?

n

1

countcompositions of n

1=1 1234

2=1+1=2 2

Proof by Crayon Tucson MS Teacher’s Circle

How many 12-compositions are there for n?

n

1

countcompositions of n

1=1 1234

2=1+1=23=1+1+1=1+2=2+1

2

Proof by Crayon Tucson MS Teacher’s Circle

How many 12-compositions are there for n?

n

1

countcompositions of n

1=1 1234

2=1+1=23=1+1+1=1+2=2+1

23

Proof by Crayon Tucson MS Teacher’s Circle

How many 12-compositions are there for n?

n

1

countcompositions of n

1=1 1234

2=1+1=23=1+1+1=1+2=2+14=1+1+1+1=1+1+2

=1+2+1=2+1+1=2+2

23

Proof by Crayon Tucson MS Teacher’s Circle

How many 12-compositions are there for n?

n

1

countcompositions of n

1=1 1234

2=1+1=23=1+1+1=1+2=2+14=1+1+1+1=1+1+2

=1+2+1=2+1+1=2+2

235

Proof by Crayon Tucson MS Teacher’s Circle

The number of ways to tile a 2xn board with horizontal or vertical dominoes

The number of ways to tile a 1xn board with squares and dominoes

The number of 12-compositions of n

All of the following are counted by Fibonacci numbers!

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