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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 545 · 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.

NameDateTime takenScore ___ / 40
  1. 1.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) 12 (c) 24 (d) 13
  2. 2.A bag has 3 white 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) 7 (b) 4 (c) 5 (d) 2
  3. 3.A drawer has lots of red, green, 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) 2 (b) 4 (c) 5 (d) 8
  4. 4.A bag has 8 red balls and 10 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 9 (b) 18 (c) 2 (d) 10
  5. 5.A bag has 10 yellow 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) 15 (c) 11 (d) 12
  6. 6.A drawer has lots of blue, black, 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) 4 (b) 2 (c) 5 (d) 8
  7. 7.A bag has 8 green balls and 9 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 9 (b) 2 (c) 10 (d) 17
  8. 8.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) 14 (c) 22 (d) 21
  9. 9.A bag has 8 red 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) 8 (b) 15 (c) 9 (d) 2
  10. 10.A drawer has lots of blue, 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) 6 (c) 4 (d) 3
  11. 11.A bag has 7 red balls and 10 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 8 (b) 17 (c) 9 (d) 2
  12. 12.A bag has 6 white balls and 4 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 8 (b) 2 (c) 10 (d) 7
  13. 13.A bag has 12 yellow 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) 13 (b) 17 (c) 2 (d) 6
  14. 14.A bag has 3 green balls and 10 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) 13 (d) 11
  15. 15.A drawer has lots of red, 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) 4 (d) 6
  16. 16.A bag has 4 yellow balls and 7 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 11 (b) 5 (c) 2 (d) 6
  17. 17.A bag has 12 red 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) 19 (b) 8 (c) 2 (d) 13
  18. 18.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) 21 (c) 29 (d) 22
  19. 19.A bag has 8 white 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) 2 (c) 20 (d) 9
  20. 20.A bag has 11 blue balls and 8 red 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) 12 (d) 19
  21. 21.A bag has 5 blue balls and 4 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 9 (b) 2 (c) 6 (d) 7
  22. 22.A bag has 3 black balls and 7 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) 4 (d) 8
  23. 23.A bag has 10 green balls and 8 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 18 (b) 11 (c) 2 (d) 12
  24. 24.A drawer has lots of red, black, blue, 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) 6 (c) 10 (d) 5
  25. 25.A drawer has lots of blue, green, yellow, 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) 2 (c) 10 (d) 6
  26. 26.A drawer has lots of white, green, 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) 5 (b) 6 (c) 2 (d) 10
  27. 27.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) 36 (b) 37 (c) 25 (d) 24
  28. 28.A bag has 4 white balls and 7 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 5 (b) 11 (c) 6 (d) 2
  29. 29.A bag has 8 yellow 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) 15 (b) 8 (c) 9 (d) 2
  30. 30.A bag has 11 black 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) 19 (b) 9 (c) 12 (d) 2
  31. 31.A drawer has lots of 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) 2 (b) 6 (c) 3 (d) 4
  32. 32.A bag has 3 red balls and 7 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) 8 (c) 2 (d) 10
  33. 33.A drawer has lots of black, white, 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) 4 (b) 2 (c) 8 (d) 5
  34. 34.A bag has 4 green balls and 3 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 2 (b) 7 (c) 6 (d) 5
  35. 35.A bag has 10 blue 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) 11 (b) 4 (c) 2 (d) 13
  36. 36.A bag has 5 yellow 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) 8 (b) 2 (c) 6 (d) 4
  37. 37.A bag has 11 red 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) 15 (b) 5 (c) 12 (d) 2
  38. 38.A drawer has lots of 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) 6 (b) 3 (c) 4 (d) 2
  39. 39.A bag has 6 red balls and 11 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 7 (b) 17 (c) 2 (d) 8
  40. 40.A drawer has lots of red, blue, yellow, 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) 10 (c) 5 (d) 6

Answer key

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

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