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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 411 · 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.

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

Answer key

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

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