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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 302 · 40 questions
TalentJR

How to solve it (Think of the unluckiest case): Imagine the unluckiest draws: how many can you take and still NOT have what you want? Add 1 to that number — the next one must give it. Pigeonhole: with n boxes (colours, months, days), n × (k − 1) + 1 things make sure some box has at least k.

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  1. 1.A bag has 5 red balls and 6 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 7 (b) 6 (c) 2 (d) 11
  2. 2.What is the smallest number of children in a group that makes sure at least two of them were born in the same month? (a) 12 (b) 24 (c) 13 (d) 25
  3. 3.A bag has 6 red balls and 12 black balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 2 (b) 7 (c) 18 (d) 13
  4. 4.A bag has 10 blue balls and 11 black balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 11 (b) 2 (c) 21 (d) 12
  5. 5.What is the smallest number of children in a group that makes sure at least three of them were born in the same day of the week? (a) 14 (b) 22 (c) 21 (d) 15
  6. 6.A drawer has lots of red, blue, white, black and green socks, all mixed up. In the dark, how many socks must you take out to be sure of getting a matching pair? (a) 6 (b) 5 (c) 10 (d) 2
  7. 7.A drawer has lots of yellow and green socks, all mixed up. In the dark, how many socks must you take out to be sure of getting a matching pair? (a) 5 (b) 4 (c) 2 (d) 3
  8. 8.A drawer has lots of black, blue, green, white and yellow socks, all mixed up. In the dark, how many socks must you take out to be sure of getting a matching pair? (a) 5 (b) 6 (c) 10 (d) 2
  9. 9.A bag has 9 yellow balls and 10 black balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 2 (b) 19 (c) 11 (d) 10
  10. 10.A drawer has lots of green, black, blue and red socks, all mixed up. In the dark, how many socks must you take out to be sure of getting a matching pair? (a) 8 (b) 2 (c) 5 (d) 4
  11. 11.A bag has 5 blue balls and 4 red balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 2 (b) 6 (c) 9 (d) 5
  12. 12.A bag has 11 white balls and 5 black balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 2 (b) 12 (c) 16 (d) 6
  13. 13.A drawer has lots of red, white, green, black and yellow socks, all mixed up. In the dark, how many socks must you take out to be sure of getting a matching pair? (a) 2 (b) 5 (c) 10 (d) 6
  14. 14.What is the smallest number of children in a group that makes sure at least four of them were born in the same day of the week? (a) 21 (b) 28 (c) 22 (d) 29
  15. 15.A bag has 11 yellow balls and 5 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 2 (b) 12 (c) 16 (d) 13
  16. 16.A bag has 12 white balls and 4 red balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 2 (b) 13 (c) 16 (d) 5
  17. 17.What is the smallest number of children in a group that makes sure at least two of them were born in the same day of the week? (a) 15 (b) 8 (c) 14 (d) 7
  18. 18.A bag has 6 white balls and 7 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 2 (b) 13 (c) 8 (d) 7
  19. 19.A bag has 4 yellow balls and 6 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 6 (b) 5 (c) 2 (d) 10
  20. 20.A bag has 12 red balls and 5 blue balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 6 (b) 13 (c) 17 (d) 2
  21. 21.A bag has 8 blue balls and 4 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 10 (b) 2 (c) 12 (d) 9
  22. 22.A bag has 10 white balls and 8 yellow balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 11 (b) 2 (c) 9 (d) 18
  23. 23.A bag has 8 red balls and 10 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 9 (b) 18 (c) 2 (d) 10
  24. 24.A bag has 12 blue balls and 3 green balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 13 (b) 15 (c) 2 (d) 4
  25. 25.A bag has 11 red balls and 10 green balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 21 (b) 12 (c) 11 (d) 2
  26. 26.A drawer has lots of yellow, green, blue, red and white socks, all mixed up. In the dark, how many socks must you take out to be sure of getting a matching pair? (a) 5 (b) 6 (c) 2 (d) 10
  27. 27.What is the smallest number of children in a group that makes sure at least four of them were born in the same month? (a) 36 (b) 48 (c) 37 (d) 49
  28. 28.A bag has 4 red balls and 9 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 2 (b) 5 (c) 13 (d) 6
  29. 29.A bag has 10 black balls and 11 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 12 (b) 11 (c) 21 (d) 2
  30. 30.A bag has 3 red balls and 7 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 5 (b) 4 (c) 2 (d) 10
  31. 31.A bag has 9 yellow balls and 11 white balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 20 (b) 12 (c) 10 (d) 2
  32. 32.A bag has 3 green balls and 11 blue balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 12 (b) 4 (c) 2 (d) 14
  33. 33.A drawer has lots of white, blue, red and black socks, all mixed up. In the dark, how many socks must you take out to be sure of getting a matching pair? (a) 4 (b) 8 (c) 5 (d) 2
  34. 34.A drawer has lots of green, blue, black and yellow socks, all mixed up. In the dark, how many socks must you take out to be sure of getting a matching pair? (a) 2 (b) 4 (c) 8 (d) 5
  35. 35.A bag has 5 yellow balls and 11 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 2 (b) 16 (c) 6 (d) 7
  36. 36.A bag has 11 white balls and 5 red balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 16 (b) 6 (c) 2 (d) 12
  37. 37.A bag has 8 blue balls and 3 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 10 (b) 11 (c) 2 (d) 9
  38. 38.A bag has 9 yellow balls and 9 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 11 (b) 2 (c) 18 (d) 10
  39. 39.A bag has 12 black balls and 5 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 14 (b) 2 (c) 13 (d) 17
  40. 40.A bag has 12 yellow balls and 13 green balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 2 (b) 13 (c) 25 (d) 14

Answer key

  1. 7
  2. 13
  3. 13
  4. 12
  5. 15
  6. 6
  7. 3
  8. 6
  9. 11
  10. 5
  11. 6
  12. 12
  13. 6
  14. 22
  15. 13
  16. 13
  17. 8
  18. 8
  19. 6
  20. 13
  21. 10
  22. 11
  23. 10
  24. 13
  25. 12
  26. 6
  27. 37
  28. 6
  29. 12
  30. 5
  31. 12
  32. 12
  33. 5
  34. 5
  35. 7
  36. 12
  37. 10
  38. 11
  39. 14
  40. 14

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