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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 201 · 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 drawer has lots of blue 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) 2 (b) 5 (c) 3 (d) 4
  2. 2.A bag has 3 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) 4 (b) 2 (c) 8 (d) 6
  3. 3.A drawer has lots of white, blue, black, green 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) 6 (b) 10 (c) 5 (d) 2
  4. 4.A bag has 10 white balls and 4 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 2 (b) 11 (c) 14 (d) 12
  5. 5.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) 28 (b) 21 (c) 29 (d) 22
  6. 6.A bag has 9 black balls and 11 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 10 (b) 2 (c) 11 (d) 20
  7. 7.A bag has 4 yellow balls and 6 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 10 (b) 5 (c) 2 (d) 6
  8. 8.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) 14 (b) 15 (c) 8 (d) 7
  9. 9.A bag has 6 white balls and 11 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 7 (b) 8 (c) 2 (d) 17
  10. 10.A bag has 3 white balls and 9 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) 12 (c) 4 (d) 10
  11. 11.A bag has 5 green balls and 11 yellow 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) 2 (c) 16 (d) 6
  12. 12.A bag has 10 yellow balls and 4 blue balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 14 (b) 11 (c) 2 (d) 5
  13. 13.A drawer has lots of black, green, white 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) 2 (b) 4 (c) 5 (d) 8
  14. 14.A bag has 8 blue balls and 12 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 10 (b) 2 (c) 9 (d) 20
  15. 15.A bag has 8 red balls and 6 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 9 (b) 10 (c) 14 (d) 2
  16. 16.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) 37 (b) 48 (c) 36 (d) 49
  17. 17.A bag has 4 black balls and 3 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 2 (b) 7 (c) 6 (d) 5
  18. 18.A drawer has lots of black, green and blue 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) 3 (c) 6 (d) 4
  19. 19.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
  20. 20.A bag has 11 blue balls and 3 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 13 (b) 12 (c) 2 (d) 14
  21. 21.A bag has 10 green balls and 7 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 2 (b) 12 (c) 11 (d) 17
  22. 22.What is the smallest number of children in a group that makes sure at least three of them were born in the same month? (a) 36 (b) 37 (c) 24 (d) 25
  23. 23.A bag has 5 green balls and 12 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 2 (b) 17 (c) 7 (d) 6
  24. 24.A bag has 8 blue balls and 3 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 2 (b) 11 (c) 9 (d) 10
  25. 25.A bag has 7 green balls and 4 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 8 (b) 11 (c) 2 (d) 9
  26. 26.A bag has 10 yellow balls and 10 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 2 (b) 12 (c) 20 (d) 11
  27. 27.A bag has 8 blue balls and 6 black balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 9 (b) 2 (c) 14 (d) 7
  28. 28.A drawer has lots of white, red, 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) 4 (b) 8 (c) 5 (d) 2
  29. 29.A drawer has lots of green, blue, white 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) 4 (b) 8 (c) 5 (d) 2
  30. 30.A bag has 8 blue 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) 6 (b) 13 (c) 9 (d) 2
  31. 31.A drawer has lots of green 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) 3 (c) 2 (d) 5
  32. 32.A bag has 8 blue balls and 5 yellow balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 9 (b) 2 (c) 13 (d) 6
  33. 33.A drawer has lots of green 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) 5 (d) 3
  34. 34.A bag has 12 blue balls and 12 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 24 (b) 14 (c) 2 (d) 13
  35. 35.A bag has 9 green balls and 5 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 10 (b) 11 (c) 14 (d) 2
  36. 36.A bag has 4 red balls and 7 green balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 5 (b) 11 (c) 8 (d) 2
  37. 37.A bag has 7 blue balls and 4 black balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 5 (b) 2 (c) 11 (d) 8
  38. 38.A bag has 7 white balls and 6 yellow 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) 8 (c) 7 (d) 13
  39. 39.A bag has 5 green balls and 4 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 2 (b) 9 (c) 7 (d) 6
  40. 40.A bag has 10 black balls and 3 white 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) 2 (c) 11 (d) 4

Answer key

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

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