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.
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
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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.
- MacstersUndead
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) !
Hello, atreyyu!
I haven't found a proof yet,
but let me explain the problem further.
Let = side of the 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.
Since the square is composed of "dominos", must be even.
. . And that is the only constraint.
Prove: in every possible tiling of an 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.
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.