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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 85 · 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 green 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
  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) 22 (b) 29 (c) 21 (d) 28
  3. 3.A bag has 11 black balls and 10 red 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) 21 (d) 12
  4. 4.A bag has 4 red balls and 10 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 6 (b) 14 (c) 2 (d) 5
  5. 5.A bag has 12 black balls and 9 blue 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) 13 (c) 21 (d) 2
  6. 6.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
  7. 7.A bag has 5 black balls and 7 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) 12 (c) 6 (d) 8
  8. 8.A bag has 6 black balls and 7 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 7 (b) 2 (c) 13 (d) 8
  9. 9.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) 15 (b) 14 (c) 8 (d) 7
  10. 10.A bag has 10 blue 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) 11 (b) 7 (c) 16 (d) 2
  11. 11.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) 4 (b) 5 (c) 8 (d) 2
  12. 12.A drawer has lots of red, 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) 4 (b) 2 (c) 8 (d) 5
  13. 13.A bag has 8 yellow balls and 10 green 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) 18 (c) 9 (d) 2
  14. 14.A bag has 7 red balls and 5 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 12 (b) 9 (c) 8 (d) 2
  15. 15.A drawer has lots of blue, 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) 2 (b) 5 (c) 10 (d) 6
  16. 16.A drawer has lots of white, 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) 3 (c) 4 (d) 6
  17. 17.A drawer has lots of white, yellow, 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) 10 (b) 6 (c) 2 (d) 5
  18. 18.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) 36 (c) 37 (d) 25
  19. 19.A drawer has lots of white, green, blue, 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) 5 (b) 2 (c) 10 (d) 6
  20. 20.A drawer has lots of yellow, red, 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) 10 (b) 6 (c) 2 (d) 5
  21. 21.A bag has 7 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) 10
  22. 22.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) 5 (b) 3 (c) 2 (d) 4
  23. 23.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) 4 (b) 3 (c) 5 (d) 2
  24. 24.A bag has 8 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) 11 (b) 18 (c) 9 (d) 2
  25. 25.A drawer has lots of yellow, 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) 2 (b) 6 (c) 3 (d) 4
  26. 26.A drawer has lots of red, 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) 2 (c) 4 (d) 6
  27. 27.A bag has 4 black balls and 3 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 2 (b) 5 (c) 6 (d) 7
  28. 28.A drawer has lots of black, 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) 2 (b) 4 (c) 6 (d) 3
  29. 29.A drawer has lots of yellow 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) 4 (c) 5 (d) 3
  30. 30.A drawer has lots of black, red, yellow, blue 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) 6 (c) 5 (d) 10
  31. 31.A bag has 10 green balls and 12 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 22 (b) 12 (c) 2 (d) 11
  32. 32.A drawer has lots of white, 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) 2 (b) 3 (c) 6 (d) 4
  33. 33.A bag has 11 blue balls and 10 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 2 (b) 13 (c) 21 (d) 12
  34. 34.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) 3 (b) 5 (c) 4 (d) 2
  35. 35.A bag has 5 blue balls and 8 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 13 (b) 7 (c) 2 (d) 6
  36. 36.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) 4 (d) 2
  37. 37.A bag has 4 blue 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) 7 (b) 5 (c) 4 (d) 2
  38. 38.A drawer has lots of blue, white, red, 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) 6 (b) 2 (c) 5 (d) 10
  39. 39.A bag has 10 white 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) 15 (c) 12 (d) 2
  40. 40.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) 2 (b) 5 (c) 3 (d) 4

Answer key

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

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