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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 283 · 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 drawer has lots of 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) 2 (b) 4 (c) 6 (d) 3
  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) 7 (b) 14 (c) 15 (d) 8
  3. 3.A bag has 4 yellow balls and 11 red balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 12 (b) 5 (c) 2 (d) 15
  4. 4.A drawer has lots of 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) 5 (b) 2 (c) 4 (d) 3
  5. 5.A drawer has lots of red, black, yellow, 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) 2 (c) 6 (d) 10
  6. 6.A bag has 12 red balls and 3 blue 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) 2 (c) 4 (d) 13
  7. 7.A bag has 3 black balls and 8 yellow 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) 9 (c) 4 (d) 2
  8. 8.A drawer has lots of red, 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) 3 (b) 4 (c) 6 (d) 2
  9. 9.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) 21 (c) 29 (d) 28
  10. 10.A bag has 9 green balls and 12 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 21 (b) 2 (c) 11 (d) 10
  11. 11.A bag has 3 black 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) 15 (b) 4 (c) 13 (d) 2
  12. 12.A drawer has lots of yellow, 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) 3 (c) 4 (d) 6
  13. 13.A bag has 10 yellow 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) 2 (b) 14 (c) 11 (d) 5
  14. 14.A bag has 3 red 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) 4 (c) 8 (d) 6
  15. 15.A drawer has lots of red, blue, 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) 2 (c) 6 (d) 5
  16. 16.A drawer has lots of white, black, 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) 8 (b) 4 (c) 2 (d) 5
  17. 17.A drawer has lots of blue, red, green, 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) 10 (c) 6 (d) 5
  18. 18.A bag has 5 red 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) 11 (b) 2 (c) 6 (d) 7
  19. 19.A bag has 5 green balls and 11 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 7 (b) 6 (c) 16 (d) 2
  20. 20.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) 49 (b) 37 (c) 36 (d) 48
  21. 21.A bag has 6 blue balls and 12 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 18 (b) 8 (c) 2 (d) 7
  22. 22.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) 14 (b) 22 (c) 21 (d) 15
  23. 23.A bag has 6 white 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) 2 (b) 10 (c) 7 (d) 5
  24. 24.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) 25 (b) 24 (c) 13 (d) 12
  25. 25.A bag has 8 black balls and 5 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) 9 (c) 13 (d) 6
  26. 26.A bag has 7 blue balls and 4 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 9 (b) 2 (c) 8 (d) 11
  27. 27.A bag has 12 red balls and 6 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 2 (b) 18 (c) 13 (d) 14
  28. 28.A drawer has lots of red 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) 2 (c) 3 (d) 4
  29. 29.A drawer has lots of blue, yellow, 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) 5 (c) 10 (d) 6
  30. 30.A bag has 9 red balls and 4 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 2 (b) 10 (c) 13 (d) 11
  31. 31.A drawer has lots of blue, yellow, green, 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) 10 (b) 5 (c) 6 (d) 2
  32. 32.A drawer has lots of yellow 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) 5 (d) 2
  33. 33.A drawer has lots of white, red, 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
  34. 34.A bag has 6 black balls and 10 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) 16 (c) 11 (d) 7
  35. 35.A bag has 5 red balls and 4 yellow 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) 6 (c) 9 (d) 2
  36. 36.A bag has 3 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) 11 (b) 4 (c) 9 (d) 2
  37. 37.A bag has 7 white 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) 8 (b) 15 (c) 9 (d) 2
  38. 38.A drawer has lots of blue, 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) 2 (b) 6 (c) 3 (d) 4
  39. 39.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) 4 (c) 3 (d) 2
  40. 40.A bag has 5 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) 6 (c) 4 (d) 8

Answer key

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

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