1. ## Tiled square

A square is assembled out of 2-by-1 tiles. Show that there exists a 2-by-2 square assembled out of some two tiles.

2. ## tiled square

Originally Posted by atreyyu
A square is assembled out of 2-by-1 tiles. Show that there exists a 2-by-2 square assembled out of some two tiles.
Bonjour mon ami,

Are you leaving something out or do you need to restate the question?

bjh

3. Erm, no, I need to prove that in such a square I can pick two tiles that themselves form a 2-by-2 square.

4. A square is assembled out of 2-by-1 tiles. Show that there exists a 2-by-2 square assembled out of some two tiles. "[]" will be 2x1 orientation while "=" will be 1x2 orientiation
--

The base case is using two 2x1 tiles to make a 2x2 square. (side by side. row = 2, column = 1 + 1 = 2)
[][] (2x2 square)

Now all you need to show is that for any square, you have a 2x2 square. (induction?)

For example, take the next case, using 8 = 2^3 tiles. ((EDIT//: is it possible with 4? you can get something close like 3 2x1 side by side and then 1 1x2 on the bottom))

[][][][]
[][][][] (4x4 square)

The pattern seems to be, then, that you can only construct a square with 2x1 tiles if and only if you have exactly 2^odd tiles.

hope that helps.

5. What about 6x6 square? It can be laid out properly with 18 tiles, and 18 $\displaystyle \neq$ 2^odd.
I still don't know how to tackle this.

6. Originally Posted by atreyyu
What about 6x6 square? It can be laid out properly with 18 tiles, and 18 $\displaystyle \neq$ 2^odd.
I still don't know how to tackle this.
WHY are you bringing this in, since your original post was:
"A square is assembled out of 2-by-1 tiles.
Show that there exists a 2-by-2 square assembled out of some two tiles."

7. I'm bringing this in because it is wrong, as you can build a square using 18 tiles, and 18 is not an odd power of 2.

8. Ok...not sure what you're asking or what your point is, BUT:

with these tiles (1 by 2), only even-sided squares can be built

if n = side length of squares, then number of tiles to build =
n^2 / 2 ; like your 6by6 = 6^2 / 2 = 18

odd-sided squares can be built, BUT one 1by1 square always required.

C'est pas ma faute, tabarnak (en bon canadien francais) !

9. Hello, atreyyu!

I haven't found a proof yet,
but let me explain the problem further.

A square is assembled out of 2-by-1 tiles.
Show that there exists a 2-by-2 square assembled out of some two tiles.
Let $\displaystyle n$ = side of the square.

Since the square is composed of "dominos", $\displaystyle n$ must be even.
. .
And that is the only constraint.

Prove: in every possible tiling of an $\displaystyle n\times n$ square,

. . . . . there will be a pair of tiles oriented like:
Code:
      *---*---*        *-------*
|   |   |        |       |
|   |   |   or   *-------*
|   |   |        |       |
*---*---*        *-------*

I've tried various approaches (including proof by contradiction)
. . but have had no luck so far.

10. OK I thought I was making myself clear all along:

Using the 2x1 tiles, we make a square. Of course it's length can only be an even number, and only an even number of tiles could have been used; that's a fact.
The point is to show that you can pick such a 2x2 square inside the big square (that is of any size nxn, n=even) that is formed using only two tiles.

EDIT: Soroban, thanks, that's exactly what I meant.