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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 138 · 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 8 blue balls and 5 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 13 (b) 10 (c) 2 (d) 9
  2. 2.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) 21 (b) 29 (c) 28 (d) 22
  3. 3.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) 49 (c) 36 (d) 48
  4. 4.A bag has 6 white balls and 4 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 8 (b) 10 (c) 7 (d) 2
  5. 5.A drawer has lots of red, blue, 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) 8 (b) 5 (c) 2 (d) 4
  6. 6.A bag has 11 blue 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) 2 (b) 18 (c) 8 (d) 12
  7. 7.A bag has 10 red 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) 11 (b) 2 (c) 12 (d) 21
  8. 8.A bag has 6 red balls and 4 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 10 (b) 8 (c) 7 (d) 2
  9. 9.A bag has 9 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) 8 (c) 10 (d) 16
  10. 10.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) 15 (c) 14 (d) 7
  11. 11.A bag has 6 blue balls and 4 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) 7 (d) 5
  12. 12.A drawer has lots of green, blue, 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) 10 (b) 6 (c) 2 (d) 5
  13. 13.A drawer has lots of 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) 4 (b) 5 (c) 3 (d) 2
  14. 14.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) 21 (d) 15
  15. 15.A bag has 7 white balls and 5 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 8 (b) 12 (c) 2 (d) 9
  16. 16.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) 5 (c) 2 (d) 4
  17. 17.A bag has 4 red 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) 5 (c) 15 (d) 12
  18. 18.A bag has 5 yellow balls and 9 green 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) 6 (c) 14 (d) 2
  19. 19.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) 5 (b) 3 (c) 4 (d) 2
  20. 20.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) 25 (c) 24 (d) 12
  21. 21.A bag has 7 red 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) 4 (c) 8 (d) 10
  22. 22.A drawer has lots of 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) 2 (b) 4 (c) 5 (d) 3
  23. 23.A bag has 3 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) 2 (b) 15 (c) 13 (d) 4
  24. 24.A drawer has lots of blue, green, 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) 8 (d) 2
  25. 25.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
  26. 26.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) 5 (b) 4 (c) 2 (d) 3
  27. 27.A drawer has lots of black, 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) 2 (b) 6 (c) 5 (d) 10
  28. 28.A drawer has lots of red, yellow, 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) 5 (b) 10 (c) 2 (d) 6
  29. 29.A drawer has lots of black, blue, 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) 4 (b) 5 (c) 8 (d) 2
  30. 30.A bag has 4 yellow 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) 10 (c) 7 (d) 5
  31. 31.A bag has 9 red 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) 12 (d) 10
  32. 32.A bag has 7 black balls and 12 blue 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) 13 (d) 2
  33. 33.A drawer has lots of white, yellow, 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
  34. 34.A bag has 12 blue 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) 13 (b) 18 (c) 2 (d) 7
  35. 35.A bag has 6 white 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) 7 (b) 2 (c) 14 (d) 9
  36. 36.A bag has 11 blue balls and 3 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 14 (b) 12 (c) 13 (d) 2
  37. 37.A bag has 5 red balls and 9 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) 2 (c) 10 (d) 14
  38. 38.A bag has 5 yellow balls and 6 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 6 (b) 11 (c) 2 (d) 7
  39. 39.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) 25 (b) 37 (c) 24 (d) 36
  40. 40.A drawer has lots of yellow 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) 2 (b) 3 (c) 4 (d) 5

Answer key

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

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