Quantcast

A community for students. Sign up today!

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

estudier

  • 2 years 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.

  • This Question is Closed
  1. UnkleRhaukus
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    |dw:1350302056370:dw|

  2. estudier
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Yes, like that, thank u for drawing it:-)

  3. UnkleRhaukus
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    each row must have either more red, or more blue dots

  4. UnkleRhaukus
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    therefore two of the rows has four or more of a certain colour

  5. estudier
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    In total, there are 21 dots, so at least 11 are red or at least 11 are blue..

  6. UnkleRhaukus
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    i get a minimum of 8

  7. UnkleRhaukus
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    |dw:1350302644089:dw|

  8. estudier
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    OK, maybe we can assume that 11 are blue and go from there....

  9. UnkleRhaukus
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    |dw:1350302783459:dw|

  10. UnkleRhaukus
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    |dw:1350302833654:dw|

  11. estudier
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Clearly, there are some different cases to consider.....

  12. UnkleRhaukus
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    i dont know how to be general

  13. estudier
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    There is some symmetry so we could assume that the top row has 7,6,5 or 4 blue dots.

  14. sara12345
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    1 blue 20 red is also possible is it, it wont work.. just trying to understand the question

  15. estudier
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Then you are just working with reds instead of blues but the problem is still the same.

  16. sasogeek
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    is this some sort of probability question?

  17. estudier
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Um...no.

  18. sara12345
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    looks it is related to counting

  19. estudier
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Reasoning, I guess...

  20. sasogeek
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    is it necessary that the dots along the edge of the rectangle be the same colour as the dots at the vertices?

  21. estudier
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Only that 4 dots of 1 colour are the vertices..

  22. sara12345
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    |dw:1350303695070:dw|

  23. sara12345
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    atleast 2 are of red or blue dots in each column

  24. sara12345
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    |dw:1350303798138:dw|

  25. estudier
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    That's right, you may just as well stick to one colour, it is simpler......

  26. sara12345
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    in the second column only one way is possible for not to have a rectangle

  27. sasogeek
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    there's an odd number of rows and an odd number of columns, is that a clue to anything?

  28. sara12345
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    |dw:1350303894487:dw|

  29. sara12345
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    but third column we cant escape a rectangle !

  30. sara12345
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    is this ok proof ?

  31. estudier
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 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......

  32. sara12345
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    oh yeah we need to exhaust all cases

  33. sara12345
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    begin from right side

  34. sara12345
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    |dw:1350304093421:dw|

  35. estudier
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    For that case, the question becomes where are the 4 remaining dots?

  36. sara12345
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    next, to escape rectangle, we can have all 3b (or) 1r and 2 b

  37. sara12345
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    so next column we cant have a rectangle, so lets see 3rd column for each of those

  38. sara12345
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    |dw:1350304238693:dw|

  39. sara12345
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    anthing in third column would make a rectangle when 3bs are there in 2nd column

  40. sauravshakya
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Do we count square as a rectangle?

  41. estudier
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Yes.

  42. estudier
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    For the first case, we can say that the second row has at least 2 dots, right?

  43. estudier
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    For if not, they are in the third row instead....

  44. sara12345
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    or else if we just prove two exact same columns exist that would be enough

  45. estudier
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Take these 2 and the 2 above them in the first row and you get a rectangle.

  46. sasogeek
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 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

  47. estudier
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    The other 3 cases are not quite as easy but the reasoning is similar kind to that in the first case

  48. sara12345
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    i see we can have all 7 unique columns

  49. estudier
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 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....

  50. estudier
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Hmm...not much interest in this, I will close it.....

  51. MrMoose
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 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

  52. MrMoose
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    so, throughout the whole figure, you may only have 4 columns with consecutive colors: |dw:1350619522881:dw| (points arbitrarily chosen for demonstration)

  53. MrMoose
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Therefore, you must have at least 3 columns with alternating colors. These come in 2 states: R and B B R R B

  54. MrMoose
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    therefore, by the pigeonhole principle, at least 2 of the columns share the same state

  55. MrMoose
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    If two columns share the same state, they form a rectangle

  56. MrMoose
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    QED

  57. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Ask a Question
Find more explanations on OpenStudy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.