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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 270 · 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.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) 37 (c) 36 (d) 25
  2. 2.A bag has 6 white 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) 2 (b) 11 (c) 7 (d) 6
  3. 3.A bag has 5 blue 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) 5 (b) 9 (c) 2 (d) 6
  4. 4.A drawer has lots of green, 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) 5 (b) 2 (c) 4 (d) 8
  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 month? (a) 25 (b) 13 (c) 24 (d) 12
  6. 6.A drawer has lots of blue, black, 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) 2 (c) 8 (d) 5
  7. 7.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
  8. 8.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) 21 (b) 14 (c) 15 (d) 22
  9. 9.A bag has 4 white balls and 7 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) 2 (c) 11 (d) 8
  10. 10.A bag has 3 yellow balls and 10 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) 13 (c) 11 (d) 4
  11. 11.A bag has 9 red balls and 7 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 16 (b) 10 (c) 2 (d) 11
  12. 12.A bag has 6 blue 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) 7 (b) 2 (c) 16 (d) 11
  13. 13.A drawer has lots of white, 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) 6 (b) 3 (c) 4 (d) 2
  14. 14.A bag has 4 green balls and 11 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) 2 (c) 15 (d) 5
  15. 15.A bag has 4 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) 5 (b) 2 (c) 12 (d) 9
  16. 16.A bag has 8 yellow balls and 9 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 2 (b) 9 (c) 17 (d) 10
  17. 17.A bag has 6 white 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) 7 (c) 13 (d) 2
  18. 18.A bag has 5 green 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) 2 (d) 7
  19. 19.A bag has 4 red balls and 10 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 6 (b) 2 (c) 14 (d) 5
  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) 48 (b) 37 (c) 49 (d) 36
  21. 21.A drawer has lots of green, black, white, 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) 6 (b) 5 (c) 2 (d) 10
  22. 22.A bag has 4 yellow balls and 11 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) 12 (c) 15 (d) 2
  23. 23.A bag has 4 white balls and 11 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 5 (b) 2 (c) 15 (d) 6
  24. 24.A drawer has lots of green 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) 2 (c) 4 (d) 5
  25. 25.A bag has 6 green 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) 9 (b) 2 (c) 7 (d) 14
  26. 26.A bag has 7 red 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) 2 (b) 8 (c) 10 (d) 4
  27. 27.A drawer has lots of white, 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) 6 (c) 4 (d) 3
  28. 28.A bag has 10 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) 18 (b) 11 (c) 9 (d) 2
  29. 29.A drawer has lots of 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) 2 (b) 3 (c) 4 (d) 5
  30. 30.A bag has 8 black balls and 3 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) 11
  31. 31.A bag has 7 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) 8 (b) 13 (c) 2 (d) 19
  32. 32.A bag has 5 yellow 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) 6 (b) 8 (c) 2 (d) 4
  33. 33.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) 3 (b) 4 (c) 2 (d) 5
  34. 34.A drawer has lots of green, red, white, 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) 5 (b) 6 (c) 10 (d) 2
  35. 35.A drawer has lots of yellow, white, 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) 8 (b) 5 (c) 2 (d) 4
  36. 36.A bag has 9 blue 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) 2 (c) 10 (d) 15
  37. 37.A bag has 5 black balls and 10 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 6 (b) 15 (c) 2 (d) 7
  38. 38.A bag has 9 red balls and 3 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 12 (b) 11 (c) 2 (d) 10
  39. 39.A drawer has lots of 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) 4 (d) 3
  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 day of the week? (a) 28 (b) 29 (c) 22 (d) 21

Answer key

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

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