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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 193 · 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 day of the week? (a) 14 (b) 7 (c) 8 (d) 15
  2. 2.A drawer has lots of green, yellow, black, red 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) 6 (c) 10 (d) 5
  3. 3.A drawer has lots of blue, 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) 2 (b) 6 (c) 4 (d) 3
  4. 4.A bag has 4 white balls and 8 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 12 (b) 2 (c) 5 (d) 6
  5. 5.A bag has 10 white balls and 9 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 2 (b) 11 (c) 19 (d) 12
  6. 6.A drawer has lots of 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) 2 (b) 4 (c) 5 (d) 8
  7. 7.A bag has 11 black balls and 4 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 15 (b) 2 (c) 12 (d) 13
  8. 8.A bag has 3 black balls and 10 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) 4 (c) 13 (d) 2
  9. 9.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
  10. 10.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) 14 (b) 15 (c) 22 (d) 21
  11. 11.A bag has 8 red balls and 7 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 10 (b) 15 (c) 2 (d) 9
  12. 12.A bag has 5 yellow balls and 5 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 10 (b) 7 (c) 6 (d) 2
  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) 29 (b) 28 (c) 22 (d) 21
  14. 14.A bag has 8 blue balls and 8 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 9 (b) 16 (c) 2 (d) 10
  15. 15.A bag has 10 yellow balls and 4 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) 2 (c) 5 (d) 14
  16. 16.A bag has 8 green balls and 12 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 10 (b) 9 (c) 20 (d) 2
  17. 17.A drawer has lots of blue, 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) 6 (c) 4 (d) 2
  18. 18.A bag has 3 black balls and 5 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 4 (b) 8 (c) 5 (d) 2
  19. 19.A bag has 10 red balls and 10 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 20 (b) 12 (c) 2 (d) 11
  20. 20.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) 5 (b) 7 (c) 2 (d) 6
  21. 21.A bag has 8 green balls and 12 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 20 (b) 10 (c) 9 (d) 2
  22. 22.A drawer has lots of white, black 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
  23. 23.A bag has 12 red 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) 16 (c) 14 (d) 13
  24. 24.A drawer has lots of 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) 3 (b) 5 (c) 4 (d) 2
  25. 25.A drawer has lots of green, black, yellow, 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) 6 (d) 10
  26. 26.A bag has 3 blue 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) 2 (b) 4 (c) 15 (d) 13
  27. 27.A drawer has lots of yellow, black, blue, 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) 10 (b) 6 (c) 5 (d) 2
  28. 28.A bag has 9 white 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) 11 (c) 2 (d) 14
  29. 29.A bag has 6 red balls and 7 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 7 (b) 2 (c) 13 (d) 8
  30. 30.A bag has 8 green 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) 2 (b) 9 (c) 12 (d) 5
  31. 31.A bag has 9 green 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) 10 (b) 12 (c) 4 (d) 2
  32. 32.A bag has 9 yellow 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) 10 (b) 16 (c) 8 (d) 2
  33. 33.A drawer has lots of green, yellow, 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) 8 (c) 2 (d) 5
  34. 34.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) 37 (c) 49 (d) 36
  35. 35.A bag has 9 blue balls and 9 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 2 (b) 10 (c) 18 (d) 11
  36. 36.A bag has 8 yellow balls and 3 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 10 (b) 11 (c) 9 (d) 2
  37. 37.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) 5 (c) 4 (d) 3
  38. 38.A bag has 5 black balls and 8 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) 9 (c) 13 (d) 2
  39. 39.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
  40. 40.A bag has 7 blue balls and 4 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) 11 (c) 8 (d) 5

Answer key

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

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