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Pigeonhole Principle Worksheet for Maths Olympiad

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

Answer key

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

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