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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 151 · 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 4 white 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) 16 (b) 5 (c) 13 (d) 2
  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 day of the week? (a) 7 (b) 14 (c) 8 (d) 15
  3. 3.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) 13 (b) 25 (c) 12 (d) 24
  4. 4.A bag has 3 red balls and 9 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) 12 (c) 4 (d) 10
  5. 5.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) 4 (b) 6 (c) 2 (d) 3
  6. 6.A bag has 11 yellow balls and 6 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) 2 (c) 12 (d) 17
  7. 7.A drawer has lots of red, green, black, 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) 2 (b) 5 (c) 10 (d) 6
  8. 8.A bag has 11 green balls and 4 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 2 (b) 12 (c) 13 (d) 15
  9. 9.A bag has 9 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) 11 (c) 10 (d) 15
  10. 10.A bag has 8 yellow 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) 12 (b) 5 (c) 2 (d) 9
  11. 11.A bag has 7 green balls and 6 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 13 (b) 8 (c) 9 (d) 2
  12. 12.A drawer has lots of yellow, red, 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) 2 (b) 5 (c) 8 (d) 4
  13. 13.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) 28 (c) 22 (d) 21
  14. 14.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) 2 (b) 4 (c) 5 (d) 3
  15. 15.A bag has 12 black 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) 17 (b) 13 (c) 2 (d) 6
  16. 16.A bag has 3 white balls and 4 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 7 (b) 4 (c) 2 (d) 5
  17. 17.A drawer has lots of white, red, green, 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) 10 (b) 2 (c) 5 (d) 6
  18. 18.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) 3 (b) 2 (c) 4 (d) 6
  19. 19.A bag has 12 white balls and 13 red 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) 25 (c) 2 (d) 13
  20. 20.A bag has 11 white 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) 2 (b) 16 (c) 6 (d) 12
  21. 21.A bag has 5 black balls and 6 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 2 (b) 6 (c) 11 (d) 7
  22. 22.A drawer has lots of yellow, 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) 3 (b) 4 (c) 2 (d) 6
  23. 23.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
  24. 24.A drawer has lots of yellow 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) 5 (c) 4 (d) 2
  25. 25.A bag has 5 blue balls and 12 red 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) 17 (d) 6
  26. 26.A bag has 10 blue balls and 9 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) 10 (c) 11 (d) 19
  27. 27.A bag has 4 green 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) 5 (b) 14 (c) 2 (d) 11
  28. 28.A bag has 5 red balls and 3 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 6 (b) 2 (c) 8 (d) 7
  29. 29.A drawer has lots of 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) 4 (b) 5 (c) 3 (d) 2
  30. 30.A drawer has lots of white, 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) 6 (b) 4 (c) 3 (d) 2
  31. 31.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) 2 (c) 5 (d) 3
  32. 32.A bag has 8 red 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) 2 (b) 5 (c) 9 (d) 12
  33. 33.A drawer has lots of black, 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) 2 (c) 4 (d) 8
  34. 34.A bag has 3 blue balls and 6 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) 4 (c) 2 (d) 9
  35. 35.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
  36. 36.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) 2 (b) 14 (c) 9 (d) 7
  37. 37.A bag has 12 green balls and 7 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 13 (b) 19 (c) 2 (d) 14
  38. 38.A bag has 6 blue balls and 3 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 7 (b) 2 (c) 9 (d) 8
  39. 39.A drawer has lots of red, black, 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) 5 (c) 2 (d) 8
  40. 40.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) 48 (b) 37 (c) 36 (d) 49

Answer key

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

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