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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 680 · 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.

NameDateTime takenScore ___ / 40
  1. 1.A bag has 8 red 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) 2 (b) 5 (c) 12 (d) 9
  2. 2.A drawer has lots of red, white 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) 4 (b) 2 (c) 3 (d) 6
  3. 3.A drawer has lots of 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) 2 (b) 3 (c) 5 (d) 4
  4. 4.A drawer has lots of white, green, black, yellow 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) 10 (c) 6 (d) 5
  5. 5.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) 7 (c) 8 (d) 15
  6. 6.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) 25 (c) 13 (d) 24
  7. 7.A bag has 6 blue balls and 7 red balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 7 (b) 8 (c) 2 (d) 13
  8. 8.A bag has 5 red balls and 6 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 11 (b) 7 (c) 6 (d) 2
  9. 9.A bag has 4 green balls and 12 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 2 (b) 16 (c) 5 (d) 6
  10. 10.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) 49 (b) 36 (c) 48 (d) 37
  11. 11.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) 9 (b) 2 (c) 12 (d) 10
  12. 12.A bag has 8 blue balls and 4 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) 5 (c) 9 (d) 12
  13. 13.A bag has 7 yellow balls and 3 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 8 (b) 2 (c) 10 (d) 9
  14. 14.A bag has 6 red balls and 8 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 14 (b) 2 (c) 7 (d) 8
  15. 15.A drawer has lots of red, green, black 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) 4 (b) 2 (c) 5 (d) 8
  16. 16.A bag has 12 black balls and 10 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 22 (b) 2 (c) 13 (d) 14
  17. 17.A bag has 12 blue balls and 8 white 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) 9 (c) 13 (d) 20
  18. 18.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) 24 (c) 25 (d) 37
  19. 19.A bag has 3 green balls and 8 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 11 (b) 2 (c) 4 (d) 5
  20. 20.A bag has 11 red balls and 8 white balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 19 (b) 12 (c) 2 (d) 9
  21. 21.A bag has 12 black balls and 8 white 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) 20
  22. 22.A drawer has lots of green, 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) 6 (b) 3 (c) 2 (d) 4
  23. 23.A drawer has lots of 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) 6 (c) 2 (d) 3
  24. 24.A bag has 8 green balls and 10 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 9 (b) 10 (c) 2 (d) 18
  25. 25.A bag has 6 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) 15 (c) 7 (d) 10
  26. 26.A drawer has lots of 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) 3 (d) 5
  27. 27.A drawer has lots of red, black, yellow, 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) 10 (b) 6 (c) 5 (d) 2
  28. 28.A bag has 3 black balls and 10 blue 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) 4 (c) 11 (d) 2
  29. 29.A drawer has lots of green, red, black 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) 5 (b) 2 (c) 8 (d) 4
  30. 30.A bag has 10 blue balls and 5 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 2 (b) 15 (c) 11 (d) 12
  31. 31.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) 22 (b) 14 (c) 15 (d) 21
  32. 32.A bag has 6 red balls and 4 white 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) 10 (d) 7
  33. 33.A bag has 4 yellow balls and 10 white 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) 14 (d) 2
  34. 34.A drawer has lots of red, 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) 6 (b) 4 (c) 3 (d) 2
  35. 35.A bag has 5 yellow 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) 2 (b) 11 (c) 7 (d) 6
  36. 36.A bag has 5 red balls and 3 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 6 (b) 8 (c) 7 (d) 2
  37. 37.A bag has 11 white balls and 5 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 2 (b) 16 (c) 13 (d) 12
  38. 38.A bag has 5 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) 6 (b) 9 (c) 2 (d) 5
  39. 39.A drawer has lots of yellow, black, red 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) 5 (c) 4 (d) 8
  40. 40.A drawer has lots of white, blue, 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) 4 (b) 8 (c) 2 (d) 5

Answer key

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

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