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Pigeonhole Principle Worksheet for Maths Olympiad

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

Answer key

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

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