Bdmo National 2014: Junior 6

[Tahmid]

Is it possible to completely cover a $14*14$ grid by "T" shaped blocks from the diagram such that no block overlaps any other bolcks? Explain your answer with logic.

[Tahmid]

I have solved the problem in this way ,

If we colour $14*14$ grid as a cheese board then there will be same number of black and white part.
there are $14*14=196$ parts in the grid and a "T" shaped block has 4 parts.
so, if we want to fill up the $14*14$ grid with "T" shaped block we need $\frac{196}{4}=49$ blocks.

now, if we put a "T" shaped block in any place on the grid either it will cover 3 white parts and 1 black part or 3 black parts and 1 white part.
let there are m blocks which cover 3 white parts and 1 black part.
and n blocks which cover 3 black parts and 1 white part

that means, $m+n=49$ [because totally we need 49 blocks]
m blocks can cover $m[(3)W+(1)B]$ parts
n blocks can cover $n[(3)B+(1)W]$ parts
[here (i)W means there is i numbers of white parts and (j)B means there is j numbers of black parts]
total parts $m{(3)W+(1)B}+n[(3)B+(1)W]$
$\Leftrightarrow (3m)W+(m)B+(3n)B+(m)W$
$\Leftrightarrow (3m+n)W+(3n+m)B$
as there are same number of black and white parts so, $(3m+n)=(3n+m)\Leftrightarrow m=n$

now we have two equations, (1).....$m+n=49$ and (2).....$m=n$
these two equations imply that there are no integer solution of $(m,n)$.
so finally it is impossible to fill up $14*14$ grid with "T" shaped blocks

have anyone another way to solve???

[Labib]

Hi Tahmid!
Just for the sake of avoiding confusion, could you add the year (2014) to this title?
That way, if we make a post on problem sets of this year, we can refer to this problem instead of creating a new thread for it.
Another thing, Could you post the solution in English?
There is a forum rule that you have to post in English in the Olympiad section. (Rule)
[Moderator Edit: Actually, there is a little clarification about that- http://www.matholympiad.org.bd/forum/vi ... 988#p14988 ]
I would not ask you to, but this is a very nice proof ( the person who graded it, loved it ) and everyone (that includes non-bengalis) deserve to know what you did here.

[Tahmid]

.......

[Tahmid]

@labib vaia, it's not kiriti. i am tahmid
and i have wrote[actually tried ] a new solution in english. but i am weak in english . so sorry for my poor english.

[Labib]

Haha, my bad. Guess I was reading multiple threads and didn't notice whose post I was writing in. Fixed anyway.
Don't worry about the English part, I've gone through it and looks just fine. Everyone has to get started at some point. Better early than late.

[Kiriti]

At first we claim that, the \(14*14\) grid can be covered by "T" shaped blocks.

Now, we colour \(14*14\) grid like chessboard with Black and White colour. So, there are \(\frac {(14*14)}{2} = 98\) Black square and \(98\) White square.

Now, If I put a block at anywhere in the grid it will cover \(3\) White square and \(1\) Black square or \(1\) white square and \(3\) Black square. Now, \(m\) and \(n\) be two positive integer where \(m\) is the number of "T" shaped block which cover \(3\) White square and \(1\) Black square and \(n\) be the number of "T" shaped block which cover \(1\) White square and \(3\) Black square.

Where, \(3m+n = 98 = 3n+m\)
\(\Rightarrow 3m +n=3n+m\)
\(\Rightarrow m=n\)
So, we can say that. \(3m + m = 98\)
\(\Rightarrow 4m = 98\)
\(\Rightarrow m = \frac {98}{4}\)
\(\Rightarrow m = 24.5\)

But \(m\) can't be a fraction. So, we can't cover the \(14*14\) grid with these "T" shaped blocks.