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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 900 · 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 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) 2 (b) 12 (c) 19 (d) 9
  2. 2.A drawer has lots of red, black, 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) 8 (b) 4 (c) 5 (d) 2
  3. 3.A bag has 12 red balls and 4 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 2 (b) 14 (c) 13 (d) 16
  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 month? (a) 13 (b) 12 (c) 24 (d) 25
  5. 5.A bag has 5 black balls and 8 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) 6 (d) 9
  6. 6.A bag has 12 red balls and 9 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 14 (b) 13 (c) 21 (d) 2
  7. 7.A drawer has lots of white, black 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) 3 (b) 6 (c) 4 (d) 2
  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) 8 (b) 7 (c) 14 (d) 15
  9. 9.A drawer has lots of 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) 5 (b) 2 (c) 3 (d) 4
  10. 10.A bag has 12 green balls and 10 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 13 (b) 22 (c) 2 (d) 14
  11. 11.A bag has 4 black balls and 9 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) 13 (c) 5 (d) 10
  12. 12.A drawer has lots of yellow, 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) 2 (b) 4 (c) 6 (d) 3
  13. 13.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) 25 (d) 24
  14. 14.A bag has 6 red 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) 2 (b) 17 (c) 7 (d) 12
  15. 15.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) 15 (c) 14 (d) 21
  16. 16.A bag has 5 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) 9 (c) 2 (d) 6
  17. 17.A drawer has lots of red 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) 5 (c) 2 (d) 3
  18. 18.A bag has 7 red 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) 8 (c) 18 (d) 9
  19. 19.A bag has 5 white 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) 7 (c) 11 (d) 6
  20. 20.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) 5 (c) 4 (d) 2
  21. 21.A bag has 3 yellow balls and 12 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 4 (b) 5 (c) 2 (d) 15
  22. 22.A bag has 4 yellow balls and 12 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 6 (b) 5 (c) 2 (d) 16
  23. 23.A bag has 5 white balls and 9 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 2 (b) 14 (c) 6 (d) 7
  24. 24.A bag has 8 black 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) 9 (b) 2 (c) 17 (d) 10
  25. 25.A bag has 7 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) 8 (b) 2 (c) 11 (d) 5
  26. 26.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) 4 (b) 5 (c) 2 (d) 3
  27. 27.A bag has 12 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) 19 (d) 8
  28. 28.A drawer has lots of white, red, 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) 8 (b) 2 (c) 5 (d) 4
  29. 29.A drawer has lots of blue, white, 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) 10 (b) 6 (c) 5 (d) 2
  30. 30.A bag has 12 blue balls and 4 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 13 (b) 16 (c) 2 (d) 14
  31. 31.A bag has 7 green 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) 8 (b) 12 (c) 2 (d) 6
  32. 32.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) 5 (b) 2 (c) 4 (d) 3
  33. 33.A drawer has lots of 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) 2 (b) 5 (c) 3 (d) 4
  34. 34.A bag has 4 yellow balls and 5 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 6 (b) 5 (c) 2 (d) 9
  35. 35.A drawer has lots of 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) 6 (c) 4 (d) 3
  36. 36.A drawer has lots of white, 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) 3 (b) 4 (c) 6 (d) 2
  37. 37.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) 2 (b) 4 (c) 5 (d) 3
  38. 38.A bag has 3 black balls and 3 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 4 (b) 2 (c) 5 (d) 6
  39. 39.A bag has 12 black balls and 8 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 20 (b) 2 (c) 14 (d) 13
  40. 40.A bag has 7 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) 10 (b) 8 (c) 16 (d) 2

Answer key

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

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