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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 701 · 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 red, 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) 3 (c) 2 (d) 6
  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) 15 (b) 14 (c) 8 (d) 7
  3. 3.A bag has 7 black balls and 4 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 8 (b) 11 (c) 9 (d) 2
  4. 4.A bag has 5 yellow 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) 2 (b) 6 (c) 7 (d) 11
  5. 5.A drawer has lots of 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) 4 (b) 3 (c) 5 (d) 2
  6. 6.A bag has 3 black balls and 10 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 5 (b) 4 (c) 13 (d) 2
  7. 7.A drawer has lots of black, 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) 3 (b) 4 (c) 2 (d) 6
  8. 8.A bag has 8 blue balls and 9 white 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) 2 (c) 17 (d) 10
  9. 9.A bag has 4 white balls and 3 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 7 (b) 6 (c) 5 (d) 2
  10. 10.A bag has 4 black balls and 11 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 5 (b) 6 (c) 15 (d) 2
  11. 11.A drawer has lots of red, yellow, green, black 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) 10 (c) 6 (d) 5
  12. 12.A bag has 3 blue balls and 12 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 4 (b) 15 (c) 2 (d) 5
  13. 13.A bag has 12 blue balls and 7 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 13 (b) 19 (c) 2 (d) 14
  14. 14.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
  15. 15.A bag has 4 green balls and 5 black 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) 9 (c) 2 (d) 6
  16. 16.A drawer has lots of white, yellow, 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) 2 (b) 10 (c) 5 (d) 6
  17. 17.A bag has 5 green balls and 8 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) 13 (d) 9
  18. 18.A bag has 12 black 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) 6 (b) 2 (c) 17 (d) 13
  19. 19.A drawer has lots of blue, white, green, 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) 5 (b) 2 (c) 6 (d) 10
  20. 20.A bag has 3 black balls and 3 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 4 (b) 6 (c) 2 (d) 5
  21. 21.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) 28 (b) 22 (c) 29 (d) 21
  22. 22.A bag has 4 green balls and 3 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 6 (b) 7 (c) 5 (d) 2
  23. 23.A bag has 5 blue balls and 9 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) 10 (c) 14 (d) 6
  24. 24.A bag has 6 red balls and 4 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 8 (b) 2 (c) 7 (d) 10
  25. 25.A drawer has lots of yellow, blue, green, black 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) 10 (b) 5 (c) 6 (d) 2
  26. 26.A bag has 11 blue balls and 8 black balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 19 (b) 2 (c) 12 (d) 9
  27. 27.A bag has 5 white 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) 6 (b) 16 (c) 2 (d) 12
  28. 28.A drawer has lots of blue, black, white, yellow 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) 2 (c) 5 (d) 6
  29. 29.A bag has 9 blue balls and 5 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 11 (b) 2 (c) 14 (d) 10
  30. 30.A bag has 3 black balls and 4 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 2 (b) 7 (c) 4 (d) 5
  31. 31.A drawer has lots of 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) 5 (b) 3 (c) 4 (d) 2
  32. 32.A bag has 12 blue balls and 9 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 21 (b) 13 (c) 2 (d) 14
  33. 33.A drawer has lots of red, yellow, 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) 8 (d) 2
  34. 34.A bag has 10 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) 11 (b) 8 (c) 17 (d) 2
  35. 35.A drawer has lots of blue, black, 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) 5 (b) 2 (c) 4 (d) 8
  36. 36.A bag has 10 blue 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) 2 (b) 16 (c) 11 (d) 7
  37. 37.A bag has 6 white balls and 9 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 8 (b) 15 (c) 7 (d) 2
  38. 38.A drawer has lots of red, 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) 2 (b) 3 (c) 6 (d) 4
  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) 36 (c) 37 (d) 49
  40. 40.A bag has 4 red 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) 14 (c) 11 (d) 5

Answer key

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

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