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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 413 · 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 10 red 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) 7 (b) 16 (c) 2 (d) 11
  2. 2.A drawer has lots of white 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) 3 (c) 5 (d) 2
  3. 3.A bag has 4 red balls and 5 white 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) 9 (c) 5 (d) 2
  4. 4.A bag has 9 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) 6 (c) 14 (d) 10
  5. 5.A bag has 3 yellow balls and 12 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 4 (b) 2 (c) 15 (d) 5
  6. 6.A bag has 9 black balls and 4 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 10 (b) 13 (c) 2 (d) 11
  7. 7.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) 28 (c) 21 (d) 29
  8. 8.A drawer has lots of black, green, 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) 2 (b) 8 (c) 5 (d) 4
  9. 9.A bag has 12 blue balls and 10 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 14 (b) 2 (c) 22 (d) 13
  10. 10.A bag has 6 white balls and 5 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 7 (b) 11 (c) 2 (d) 8
  11. 11.A bag has 3 blue balls and 5 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) 6 (c) 8 (d) 4
  12. 12.A drawer has lots of red, 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) 6 (b) 2 (c) 3 (d) 4
  13. 13.A bag has 3 black balls and 6 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 5 (b) 2 (c) 4 (d) 9
  14. 14.A bag has 8 yellow balls and 5 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 10 (b) 13 (c) 2 (d) 9
  15. 15.A bag has 10 yellow balls and 6 white 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) 7 (d) 11
  16. 16.A bag has 12 white balls and 5 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 2 (b) 13 (c) 17 (d) 14
  17. 17.A drawer has lots of blue, white 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) 3 (c) 4 (d) 2
  18. 18.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) 3 (b) 4 (c) 2 (d) 5
  19. 19.A bag has 10 black balls and 11 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) 21 (c) 12 (d) 2
  20. 20.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) 2 (b) 3 (c) 5 (d) 4
  21. 21.A drawer has lots of blue, 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) 2 (b) 4 (c) 8 (d) 5
  22. 22.A bag has 10 blue balls and 4 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 2 (b) 11 (c) 14 (d) 12
  23. 23.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
  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) 24 (b) 12 (c) 13 (d) 25
  25. 25.A bag has 3 green balls and 9 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 12 (b) 4 (c) 5 (d) 2
  26. 26.A bag has 12 blue 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) 23 (b) 2 (c) 12 (d) 13
  27. 27.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) 10 (b) 2 (c) 5 (d) 6
  28. 28.A drawer has lots of green, white, 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) 4 (b) 2 (c) 8 (d) 5
  29. 29.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) 2 (b) 3 (c) 5 (d) 4
  30. 30.A drawer has lots of black, white, blue 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) 8 (c) 5 (d) 2
  31. 31.A drawer has lots of white, 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) 8 (b) 4 (c) 2 (d) 5
  32. 32.A drawer has lots of green, red, blue, 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) 5 (c) 2 (d) 6
  33. 33.A bag has 11 green balls and 4 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 13 (b) 2 (c) 12 (d) 15
  34. 34.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) 3 (b) 6 (c) 2 (d) 4
  35. 35.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) 15 (b) 21 (c) 14 (d) 22
  36. 36.A bag has 10 black balls and 4 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) 2 (c) 5 (d) 11
  37. 37.A bag has 8 green balls and 3 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 9 (b) 2 (c) 10 (d) 11
  38. 38.A bag has 8 blue 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) 4 (b) 9 (c) 2 (d) 11
  39. 39.A bag has 4 yellow balls and 7 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 6 (b) 2 (c) 5 (d) 11
  40. 40.A drawer has lots of white, blue, 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) 5 (b) 6 (c) 10 (d) 2

Answer key

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

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