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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 350 · 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.A bag has 10 black balls and 6 red balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 16 (b) 7 (c) 2 (d) 11
  2. 2.A drawer has lots of blue, red, green, white 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
  3. 3.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) 12 (b) 13 (c) 25 (d) 24
  4. 4.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
  5. 5.A bag has 4 red balls and 3 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 7 (b) 6 (c) 2 (d) 5
  6. 6.A drawer has lots of red 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) 2 (b) 4 (c) 5 (d) 3
  7. 7.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) 22 (b) 29 (c) 21 (d) 28
  8. 8.A drawer has lots of green, 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) 2 (b) 5 (c) 4 (d) 8
  9. 9.A bag has 9 black balls and 4 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 10 (b) 11 (c) 2 (d) 13
  10. 10.A drawer has lots of 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) 5 (b) 3 (c) 2 (d) 4
  11. 11.A drawer has lots of blue, red, 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) 6 (c) 2 (d) 5
  12. 12.A bag has 11 red 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) 12 (b) 19 (c) 9 (d) 2
  13. 13.A bag has 10 white balls and 6 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 11 (b) 16 (c) 12 (d) 2
  14. 14.A bag has 7 red balls and 8 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 8 (b) 2 (c) 9 (d) 15
  15. 15.A bag has 11 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) 5 (b) 12 (c) 15 (d) 2
  16. 16.A bag has 7 white balls and 4 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 2 (b) 11 (c) 9 (d) 8
  17. 17.A bag has 7 red balls and 12 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 8 (b) 2 (c) 9 (d) 19
  18. 18.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) 15 (c) 8 (d) 7
  19. 19.A drawer has lots of blue, 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) 3 (c) 6 (d) 4
  20. 20.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) 25 (b) 37 (c) 36 (d) 24
  21. 21.A drawer has lots of white, red, 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) 5 (b) 8 (c) 2 (d) 4
  22. 22.A drawer has lots of yellow, 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) 8 (b) 4 (c) 2 (d) 5
  23. 23.A drawer has lots of yellow, green, white 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) 8 (b) 4 (c) 5 (d) 2
  24. 24.A drawer has lots of green, white, 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) 4 (b) 8 (c) 5 (d) 2
  25. 25.A drawer has lots of yellow, 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) 3 (b) 2 (c) 4 (d) 6
  26. 26.A drawer has lots of white, green, red 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) 2 (c) 8 (d) 5
  27. 27.A drawer has lots of green, 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) 3 (b) 2 (c) 4 (d) 6
  28. 28.A drawer has lots of green, red, 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) 10 (b) 2 (c) 6 (d) 5
  29. 29.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) 2 (c) 4 (d) 5
  30. 30.A drawer has lots of white, green, 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) 2 (b) 5 (c) 10 (d) 6
  31. 31.A bag has 3 black 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) 4 (b) 9 (c) 7 (d) 2
  32. 32.A bag has 10 red balls and 11 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 21 (b) 12 (c) 2 (d) 11
  33. 33.A bag has 12 red balls and 5 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) 17 (c) 6 (d) 13
  34. 34.A bag has 3 green balls and 11 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) 4 (c) 12 (d) 14
  35. 35.A bag has 10 blue balls and 11 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) 21 (c) 2 (d) 12
  36. 36.A drawer has lots of black, blue, red 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) 4 (c) 2 (d) 5
  37. 37.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) 48 (c) 36 (d) 49
  38. 38.A drawer has lots of blue, yellow, red 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) 5 (d) 2
  39. 39.A bag has 8 red 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) 13 (b) 6 (c) 9 (d) 2
  40. 40.A bag has 4 red 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) 12 (b) 9 (c) 2 (d) 5

Answer key

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

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