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Tomas.A

  • 2 years ago

How to prove that in group of 6 people there are at least 2 people who has same amount of friends among that group?

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  1. rld613
    • 2 years ago
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    LOL

  2. Tomas.A
    • 2 years ago
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    whats so lol

  3. rld613
    • 2 years ago
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    R u sure it cld be proved

  4. Tomas.A
    • 2 years ago
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    of course it is

  5. rld613
    • 2 years ago
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    oh i see didnt read the question correctly

  6. Tomas.A
    • 2 years ago
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    it's classical problem and i have proved it somehow but i don't remember anymore

  7. amistre64
    • 2 years ago
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    do the 6 people even have to be friends? after all, a group of 6 strangers is still a group of 6

  8. Tomas.A
    • 2 years ago
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    no, but if they don't have any friends there will be 6 people who have no friends

  9. amistre64
    • 2 years ago
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    if 2 people in the group are friends; then at least 2 people have 1 friend .... something like that

  10. amistre64
    • 2 years ago
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    if 3 people are friends, then a > bc, b > ac, c>ba

  11. amistre64
    • 2 years ago
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    thats at least 2 with the same amount of friends

  12. amistre64
    • 2 years ago
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    or even a>c , c>b, c>ab

  13. Tomas.A
    • 2 years ago
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    well i translated it from lithuanian but it seems it has same meaning

  14. amistre64
    • 2 years ago
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    can 2 people be not friends? if one of them is a friend?

  15. JamesJ
    • 2 years ago
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    oh wait, I see. Let f(n) be the number of friends in that group that the nth person has. For each n, \[ 0 \leq f(n) \leq 5 \] Therefore ...

  16. amistre64
    • 2 years ago
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    a>c, b>c , c>ab is what i meant lol

  17. Tomas.A
    • 2 years ago
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    yes there can be no friends at all or 1 person can have no friends

  18. amistre64
    • 2 years ago
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    I think James has it :)

  19. amistre64
    • 2 years ago
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    even if 4 are friends, that still leaves 2 with 0 friends each which is the same amount

  20. JamesJ
    • 2 years ago
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    there are 6 values for f(1), f(2), f(3), f(4), f(5), f(6). Now it can't be that these six number take on all of 0,1,2,3,4,5. Because if f(n) = 5 for one n, then f(j) > 0 for all other \( j \neq n \). Hence in fact, there are only 5 possible values for f(n). But as there are six f(n), at least two of the f(n) must be equal.

  21. amistre64
    • 2 years ago
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    ill concede to that answer as well ;)

  22. amistre64
    • 2 years ago
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    my by case proof could get lengthy

  23. amistre64
    • 2 years ago
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    gotta hang it on the left to be a valid q lol

  24. amistre64
    • 2 years ago
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    and I gotta get my ode hw written up so ciao yall :)

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