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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 135 · 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 white 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) 9 (b) 11 (c) 4 (d) 2
  2. 2.A bag has 5 white 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) 14 (b) 6 (c) 2 (d) 10
  3. 3.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) 2 (b) 5 (c) 4 (d) 3
  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) 36 (d) 49
  5. 5.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) 29 (b) 22 (c) 28 (d) 21
  6. 6.A bag has 3 red 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) 4 (c) 6 (d) 8
  7. 7.A drawer has lots of blue, 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) 4 (c) 3 (d) 2
  8. 8.A bag has 12 black 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) 2 (b) 20 (c) 9 (d) 13
  9. 9.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) 4 (b) 5 (c) 3 (d) 2
  10. 10.A drawer has lots of green, white, yellow, 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) 6 (b) 10 (c) 5 (d) 2
  11. 11.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
  12. 12.A drawer has lots of black, 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) 6 (b) 3 (c) 4 (d) 2
  13. 13.A bag has 4 yellow balls and 10 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 5 (b) 6 (c) 14 (d) 2
  14. 14.A bag has 8 red 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) 20 (b) 9 (c) 13 (d) 2
  15. 15.A bag has 12 red balls and 4 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 2 (b) 16 (c) 13 (d) 14
  16. 16.A bag has 5 yellow balls and 3 red 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) 8 (d) 6
  17. 17.A bag has 8 red 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) 13 (b) 9 (c) 2 (d) 20
  18. 18.A bag has 4 green balls and 5 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) 9 (c) 5 (d) 6
  19. 19.A bag has 5 green balls and 12 yellow 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) 2 (c) 6 (d) 17
  20. 20.A bag has 11 green 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) 2 (b) 7 (c) 17 (d) 12
  21. 21.A bag has 4 yellow balls and 5 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 9 (b) 2 (c) 5 (d) 6
  22. 22.A bag has 5 blue 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) 6 (c) 4 (d) 8
  23. 23.A bag has 8 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) 15 (b) 8 (c) 2 (d) 9
  24. 24.A drawer has lots of yellow, green, black, 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) 5 (c) 10 (d) 6
  25. 25.A drawer has lots of yellow, green 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) 6 (b) 2 (c) 3 (d) 4
  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) 3 (b) 4 (c) 2 (d) 6
  27. 27.A drawer has lots of blue, white, 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) 2 (b) 5 (c) 6 (d) 10
  28. 28.A drawer has lots of red, 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) 3 (b) 2 (c) 6 (d) 4
  29. 29.A bag has 5 white balls and 3 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 6 (b) 2 (c) 7 (d) 8
  30. 30.A drawer has lots of yellow, blue, red, black 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) 10 (b) 2 (c) 5 (d) 6
  31. 31.A bag has 9 yellow balls and 5 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 2 (b) 11 (c) 10 (d) 14
  32. 32.A bag has 9 red balls and 5 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 11 (b) 2 (c) 10 (d) 14
  33. 33.A bag has 12 yellow 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) 16 (b) 2 (c) 5 (d) 13
  34. 34.A drawer has lots of white, yellow, blue, 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) 6 (b) 2 (c) 10 (d) 5
  35. 35.A drawer has lots of black 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) 4 (b) 3 (c) 5 (d) 2
  36. 36.A bag has 4 black balls and 3 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 6 (b) 7 (c) 2 (d) 5
  37. 37.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) 36 (b) 25 (c) 37 (d) 24
  38. 38.A drawer has lots of yellow, 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) 6 (b) 10 (c) 2 (d) 5
  39. 39.A bag has 12 yellow balls and 7 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 19 (b) 13 (c) 2 (d) 14
  40. 40.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) 2 (b) 4 (c) 5 (d) 3

Answer key

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

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