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A lattice of 21 dots, some red, some blue, in 3 rows of 7.
Show that some 4 dots of 1 colour form the vertices of a rectangle.
 one year ago
 one year ago
A lattice of 21 dots, some red, some blue, in 3 rows of 7. Show that some 4 dots of 1 colour form the vertices of a rectangle.
 one year ago
 one year ago

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UnkleRhaukusBest ResponseYou've already chosen the best response.0
dw:1350302056370:dw
 one year ago

estudierBest ResponseYou've already chosen the best response.0
Yes, like that, thank u for drawing it:)
 one year ago

UnkleRhaukusBest ResponseYou've already chosen the best response.0
each row must have either more red, or more blue dots
 one year ago

UnkleRhaukusBest ResponseYou've already chosen the best response.0
therefore two of the rows has four or more of a certain colour
 one year ago

estudierBest ResponseYou've already chosen the best response.0
In total, there are 21 dots, so at least 11 are red or at least 11 are blue..
 one year ago

UnkleRhaukusBest ResponseYou've already chosen the best response.0
i get a minimum of 8
 one year ago

UnkleRhaukusBest ResponseYou've already chosen the best response.0
dw:1350302644089:dw
 one year ago

estudierBest ResponseYou've already chosen the best response.0
OK, maybe we can assume that 11 are blue and go from there....
 one year ago

UnkleRhaukusBest ResponseYou've already chosen the best response.0
dw:1350302783459:dw
 one year ago

UnkleRhaukusBest ResponseYou've already chosen the best response.0
dw:1350302833654:dw
 one year ago

estudierBest ResponseYou've already chosen the best response.0
Clearly, there are some different cases to consider.....
 one year ago

UnkleRhaukusBest ResponseYou've already chosen the best response.0
i dont know how to be general
 one year ago

estudierBest ResponseYou've already chosen the best response.0
There is some symmetry so we could assume that the top row has 7,6,5 or 4 blue dots.
 one year ago

sara12345Best ResponseYou've already chosen the best response.0
1 blue 20 red is also possible is it, it wont work.. just trying to understand the question
 one year ago

estudierBest ResponseYou've already chosen the best response.0
Then you are just working with reds instead of blues but the problem is still the same.
 one year ago

sasogeekBest ResponseYou've already chosen the best response.0
is this some sort of probability question?
 one year ago

sara12345Best ResponseYou've already chosen the best response.0
looks it is related to counting
 one year ago

estudierBest ResponseYou've already chosen the best response.0
Reasoning, I guess...
 one year ago

sasogeekBest ResponseYou've already chosen the best response.0
is it necessary that the dots along the edge of the rectangle be the same colour as the dots at the vertices?
 one year ago

estudierBest ResponseYou've already chosen the best response.0
Only that 4 dots of 1 colour are the vertices..
 one year ago

sara12345Best ResponseYou've already chosen the best response.0
dw:1350303695070:dw
 one year ago

sara12345Best ResponseYou've already chosen the best response.0
atleast 2 are of red or blue dots in each column
 one year ago

sara12345Best ResponseYou've already chosen the best response.0
dw:1350303798138:dw
 one year ago

estudierBest ResponseYou've already chosen the best response.0
That's right, you may just as well stick to one colour, it is simpler......
 one year ago

sara12345Best ResponseYou've already chosen the best response.0
in the second column only one way is possible for not to have a rectangle
 one year ago

sasogeekBest ResponseYou've already chosen the best response.0
there's an odd number of rows and an odd number of columns, is that a clue to anything?
 one year ago

sara12345Best ResponseYou've already chosen the best response.0
dw:1350303894487:dw
 one year ago

sara12345Best ResponseYou've already chosen the best response.0
but third column we cant escape a rectangle !
 one year ago

estudierBest ResponseYou've already chosen the best response.0
I think we should first start with the simplest case, which is when the complete row (say the top one) is all one colour......
 one year ago

sara12345Best ResponseYou've already chosen the best response.0
oh yeah we need to exhaust all cases
 one year ago

sara12345Best ResponseYou've already chosen the best response.0
begin from right side
 one year ago

sara12345Best ResponseYou've already chosen the best response.0
dw:1350304093421:dw
 one year ago

estudierBest ResponseYou've already chosen the best response.0
For that case, the question becomes where are the 4 remaining dots?
 one year ago

sara12345Best ResponseYou've already chosen the best response.0
next, to escape rectangle, we can have all 3b (or) 1r and 2 b
 one year ago

sara12345Best ResponseYou've already chosen the best response.0
so next column we cant have a rectangle, so lets see 3rd column for each of those
 one year ago

sara12345Best ResponseYou've already chosen the best response.0
dw:1350304238693:dw
 one year ago

sara12345Best ResponseYou've already chosen the best response.0
anthing in third column would make a rectangle when 3bs are there in 2nd column
 one year ago

sauravshakyaBest ResponseYou've already chosen the best response.0
Do we count square as a rectangle?
 one year ago

estudierBest ResponseYou've already chosen the best response.0
For the first case, we can say that the second row has at least 2 dots, right?
 one year ago

estudierBest ResponseYou've already chosen the best response.0
For if not, they are in the third row instead....
 one year ago

sara12345Best ResponseYou've already chosen the best response.0
or else if we just prove two exact same columns exist that would be enough
 one year ago

estudierBest ResponseYou've already chosen the best response.0
Take these 2 and the 2 above them in the first row and you get a rectangle.
 one year ago

sasogeekBest ResponseYou've already chosen the best response.0
what's the worst case scenario? in terms of number of dots for both colours... 11 of colour A and 10 of colour B... one colour must have even dots, and another odd dots the square would result from the colour with even dots what's the smallest rectangle we can have, what's the largest..... i'm only thinking, idk if it helps or makes any sense but it's what i can contribute to the solution xD hope it helps
 one year ago

estudierBest ResponseYou've already chosen the best response.0
The other 3 cases are not quite as easy but the reasoning is similar kind to that in the first case
 one year ago

sara12345Best ResponseYou've already chosen the best response.0
i see we can have all 7 unique columns
 one year ago

estudierBest ResponseYou've already chosen the best response.0
For the second case, you may as well assume that the first 6 are blue and then try to visualize where the remaining 5 are....
 one year ago

estudierBest ResponseYou've already chosen the best response.0
Hmm...not much interest in this, I will close it.....
 one year ago

MrMooseBest ResponseYou've already chosen the best response.1
If we take an arbitrary row of reds and blues, only 2 of the colors of the row on top of it may be the same (1 red and 1 blue), otherwise you get a rectangles: dw:1350619459441:dw leads to a rectangle
 one year ago

MrMooseBest ResponseYou've already chosen the best response.1
so, throughout the whole figure, you may only have 4 columns with consecutive colors: dw:1350619522881:dw (points arbitrarily chosen for demonstration)
 one year ago

MrMooseBest ResponseYou've already chosen the best response.1
Therefore, you must have at least 3 columns with alternating colors. These come in 2 states: R and B B R R B
 one year ago

MrMooseBest ResponseYou've already chosen the best response.1
therefore, by the pigeonhole principle, at least 2 of the columns share the same state
 one year ago

MrMooseBest ResponseYou've already chosen the best response.1
If two columns share the same state, they form a rectangle
 one year ago
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