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Pigeonhole Principle Worksheet for Maths Olympiad

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

Answer key

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

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