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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 961 · 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 5 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) 13 (b) 9 (c) 2 (d) 6
  2. 2.A bag has 5 white balls and 10 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 2 (b) 15 (c) 6 (d) 7
  3. 3.A bag has 8 yellow balls and 3 blue 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) 9 (c) 4 (d) 2
  4. 4.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) 7 (c) 14 (d) 8
  5. 5.A bag has 6 green balls and 12 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 18 (b) 7 (c) 8 (d) 2
  6. 6.A bag has 7 green balls and 6 white 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) 13 (c) 2 (d) 8
  7. 7.A bag has 8 black balls and 11 green 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) 12 (c) 19 (d) 2
  8. 8.A drawer has lots of green, red 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) 3 (b) 6 (c) 2 (d) 4
  9. 9.A bag has 11 blue balls and 10 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 2 (b) 13 (c) 21 (d) 12
  10. 10.A bag has 6 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) 13 (b) 2 (c) 7 (d) 8
  11. 11.A bag has 12 blue balls and 6 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 2 (b) 14 (c) 18 (d) 13
  12. 12.A drawer has lots of white 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) 3 (b) 2 (c) 4 (d) 5
  13. 13.A bag has 8 green 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) 2 (b) 13 (c) 6 (d) 9
  14. 14.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
  15. 15.A bag has 5 white 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) 6 (c) 8 (d) 7
  16. 16.A drawer has lots of red, blue 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) 3 (b) 2 (c) 6 (d) 4
  17. 17.A drawer has lots of blue, 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) 2 (b) 4 (c) 3 (d) 6
  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 day of the week? (a) 15 (b) 21 (c) 22 (d) 14
  19. 19.A bag has 7 yellow balls and 10 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) 11 (c) 8 (d) 17
  20. 20.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) 4 (b) 5 (c) 2 (d) 3
  21. 21.A bag has 3 blue balls and 8 green 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) 11 (c) 4 (d) 2
  22. 22.A bag has 11 black balls and 11 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 13 (b) 2 (c) 12 (d) 22
  23. 23.A drawer has lots of black, yellow, blue 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) 8 (c) 2 (d) 5
  24. 24.A bag has 3 blue 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) 4 (b) 2 (c) 7 (d) 9
  25. 25.A bag has 5 green 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) 6 (c) 7 (d) 10
  26. 26.A drawer has lots of blue, 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) 4 (c) 3 (d) 2
  27. 27.A bag has 3 red balls and 8 green 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) 9 (c) 4 (d) 2
  28. 28.A bag has 10 white 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) 2 (b) 13 (c) 4 (d) 11
  29. 29.A drawer has lots of 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) 2 (b) 5 (c) 4 (d) 3
  30. 30.A bag has 10 green balls and 7 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 17 (b) 12 (c) 2 (d) 11
  31. 31.A drawer has lots of white, red, green, blue 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) 10 (b) 2 (c) 5 (d) 6
  32. 32.A bag has 4 green balls and 10 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) 14 (c) 11 (d) 5
  33. 33.A drawer has lots of blue, 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) 3 (b) 2 (c) 4 (d) 6
  34. 34.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) 4 (d) 3
  35. 35.A bag has 4 white balls and 5 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 5 (b) 9 (c) 2 (d) 6
  36. 36.A bag has 6 green 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) 2 (b) 9 (c) 7 (d) 14
  37. 37.A drawer has lots of 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) 3 (b) 4 (c) 5 (d) 2
  38. 38.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) 29 (b) 22 (c) 28 (d) 21
  39. 39.A bag has 6 green balls and 8 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) 9 (c) 14 (d) 7
  40. 40.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) 24 (b) 36 (c) 25 (d) 37

Answer key

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

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