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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 403 · 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) 13 (b) 24 (c) 25 (d) 12
  2. 2.A drawer has lots of red, yellow, 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) 2 (b) 5 (c) 4 (d) 8
  3. 3.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) 37 (b) 36 (c) 48 (d) 49
  4. 4.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
  5. 5.A drawer has lots of white, 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) 8 (b) 2 (c) 4 (d) 5
  6. 6.A bag has 12 red balls and 13 white balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 14 (b) 13 (c) 2 (d) 25
  7. 7.A drawer has lots of blue, 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) 3 (c) 2 (d) 4
  8. 8.A bag has 4 black balls and 3 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 6 (b) 5 (c) 2 (d) 7
  9. 9.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
  10. 10.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) 6 (b) 4 (c) 3 (d) 2
  11. 11.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) 21 (d) 22
  12. 12.A bag has 4 blue balls and 6 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 2 (b) 5 (c) 10 (d) 6
  13. 13.A bag has 6 blue 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) 7 (b) 2 (c) 6 (d) 11
  14. 14.A drawer has lots of blue, red, 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) 8 (b) 2 (c) 5 (d) 4
  15. 15.A bag has 3 green 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) 4 (b) 2 (c) 7 (d) 5
  16. 16.A bag has 7 red balls and 4 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 2 (b) 8 (c) 11 (d) 9
  17. 17.A bag has 5 white balls and 3 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 6 (b) 8 (c) 2 (d) 7
  18. 18.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) 3 (c) 2 (d) 5
  19. 19.A bag has 3 blue 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) 13 (b) 11 (c) 2 (d) 4
  20. 20.A bag has 12 yellow balls and 9 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 21 (b) 13 (c) 14 (d) 2
  21. 21.A drawer has lots of black, yellow, red, 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) 6 (d) 10
  22. 22.A drawer has lots of 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) 5 (b) 2 (c) 4 (d) 3
  23. 23.A bag has 12 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) 12 (b) 23 (c) 13 (d) 2
  24. 24.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) 6 (b) 2 (c) 9 (d) 5
  25. 25.A bag has 10 red balls and 11 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 12 (b) 21 (c) 2 (d) 11
  26. 26.A bag has 4 green balls and 9 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 13 (b) 6 (c) 5 (d) 2
  27. 27.A drawer has lots of 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) 2 (b) 4 (c) 5 (d) 3
  28. 28.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) 8 (c) 7 (d) 14
  29. 29.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) 21 (c) 22 (d) 28
  30. 30.A bag has 11 red balls and 11 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 13 (b) 2 (c) 22 (d) 12
  31. 31.A drawer has lots of black, red, 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) 5 (b) 4 (c) 2 (d) 8
  32. 32.A drawer has lots of yellow, blue, black, 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) 10 (b) 2 (c) 6 (d) 5
  33. 33.A drawer has lots of yellow, white, 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) 5 (b) 4 (c) 8 (d) 2
  34. 34.A bag has 7 blue 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) 2 (b) 15 (c) 9 (d) 8
  35. 35.A drawer has lots of green, 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) 3 (c) 6 (d) 4
  36. 36.A bag has 11 green 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) 18 (b) 8 (c) 2 (d) 12
  37. 37.A bag has 4 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) 13 (b) 5 (c) 2 (d) 10
  38. 38.A bag has 11 black balls and 9 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 12 (b) 2 (c) 13 (d) 20
  39. 39.A drawer has lots of 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) 4 (b) 2 (c) 3 (d) 5
  40. 40.A drawer has lots of green, white, 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) 4 (b) 2 (c) 8 (d) 5

Answer key

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

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