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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 221 · 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, white, 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) 8 (c) 2 (d) 5
  2. 2.A bag has 12 blue balls and 6 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 2 (b) 14 (c) 18 (d) 13
  3. 3.A drawer has lots of green, 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) 4 (b) 2 (c) 8 (d) 5
  4. 4.A bag has 5 green balls and 11 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 6 (b) 7 (c) 16 (d) 2
  5. 5.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) 2 (c) 4 (d) 3
  6. 6.A drawer has lots of red, black, green, 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) 5 (b) 2 (c) 10 (d) 6
  7. 7.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) 13 (c) 12 (d) 25
  8. 8.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) 5 (b) 4 (c) 2 (d) 3
  9. 9.A bag has 5 blue 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) 11 (b) 7 (c) 6 (d) 2
  10. 10.A bag has 12 blue 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) 2 (b) 13 (c) 16 (d) 5
  11. 11.A drawer has lots of white, 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
  12. 12.A bag has 3 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) 7 (c) 5 (d) 4
  13. 13.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) 21 (b) 28 (c) 29 (d) 22
  14. 14.A bag has 4 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) 5 (c) 14 (d) 6
  15. 15.A bag has 9 red balls and 3 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 12 (b) 2 (c) 10 (d) 11
  16. 16.A drawer has lots of 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) 2 (c) 4 (d) 5
  17. 17.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
  18. 18.A bag has 6 white balls and 10 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 2 (b) 7 (c) 16 (d) 8
  19. 19.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
  20. 20.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) 15 (c) 14 (d) 8
  21. 21.A bag has 10 white 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) 11 (b) 5 (c) 14 (d) 2
  22. 22.A bag has 3 yellow balls and 10 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 13 (b) 5 (c) 4 (d) 2
  23. 23.A drawer has lots of green, blue, 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) 5 (b) 6 (c) 10 (d) 2
  24. 24.A drawer has lots of 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) 4 (b) 2 (c) 5 (d) 3
  25. 25.A bag has 8 yellow balls and 6 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 14 (b) 9 (c) 2 (d) 10
  26. 26.A drawer has lots of 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) 4 (b) 5 (c) 2 (d) 3
  27. 27.A drawer has lots of black, red, 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) 10 (b) 6 (c) 2 (d) 5
  28. 28.A bag has 10 blue balls and 8 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 2 (b) 11 (c) 18 (d) 12
  29. 29.A bag has 9 white balls and 7 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 16 (b) 2 (c) 11 (d) 10
  30. 30.A drawer has lots of yellow, black, white 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) 5 (c) 8 (d) 4
  31. 31.A bag has 8 blue balls and 8 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 9 (b) 2 (c) 10 (d) 16
  32. 32.A bag has 7 black balls and 9 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 8 (b) 9 (c) 16 (d) 2
  33. 33.A drawer has lots of black, 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) 2 (b) 6 (c) 3 (d) 4
  34. 34.A bag has 4 blue balls and 5 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 5 (b) 6 (c) 9 (d) 2
  35. 35.A bag has 12 black 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) 14 (c) 13 (d) 15
  36. 36.A drawer has lots of green, red, black, 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) 5 (b) 2 (c) 6 (d) 10
  37. 37.A bag has 11 red balls and 8 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 12 (b) 13 (c) 2 (d) 19
  38. 38.A bag has 5 black balls and 6 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 7 (b) 2 (c) 6 (d) 11
  39. 39.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
  40. 40.A bag has 7 yellow balls and 3 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) 10 (c) 2 (d) 8

Answer key

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

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