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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 442 · 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 5 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) 6 (b) 5 (c) 9 (d) 2
  2. 2.A bag has 3 white balls and 8 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 2 (b) 11 (c) 5 (d) 4
  3. 3.A bag has 4 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) 6 (b) 5 (c) 9 (d) 2
  4. 4.A bag has 11 blue 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) 23 (b) 2 (c) 12 (d) 13
  5. 5.A bag has 9 red balls and 3 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 12 (b) 11 (c) 10 (d) 2
  6. 6.A drawer has lots of green, 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) 3 (b) 6 (c) 2 (d) 4
  7. 7.A bag has 10 white balls and 6 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) 11 (c) 7 (d) 16
  8. 8.A drawer has lots of white, 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) 2 (c) 10 (d) 6
  9. 9.A bag has 4 white 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) 2 (b) 4 (c) 7 (d) 5
  10. 10.A bag has 10 black balls and 12 green 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) 11 (d) 22
  11. 11.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) 49 (d) 37
  12. 12.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) 4 (b) 2 (c) 3 (d) 5
  13. 13.A bag has 9 green balls and 11 red 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) 20 (c) 10 (d) 2
  14. 14.A bag has 8 blue 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) 7 (b) 14 (c) 2 (d) 9
  15. 15.A bag has 6 green 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) 2 (b) 18 (c) 13 (d) 7
  16. 16.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) 4 (c) 5 (d) 3
  17. 17.A bag has 7 red balls and 12 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 9 (b) 19 (c) 8 (d) 2
  18. 18.A bag has 12 black 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) 14 (c) 16 (d) 13
  19. 19.A bag has 9 green balls and 9 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 10 (b) 2 (c) 11 (d) 18
  20. 20.A drawer has lots of black, 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) 6 (c) 3 (d) 2
  21. 21.A bag has 3 white balls and 4 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 4 (b) 5 (c) 2 (d) 7
  22. 22.A drawer has lots of red, blue, 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) 4 (b) 8 (c) 2 (d) 5
  23. 23.A bag has 9 yellow balls and 3 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) 10 (c) 2 (d) 4
  24. 24.A drawer has lots of 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) 5 (b) 3 (c) 2 (d) 4
  25. 25.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) 3 (b) 5 (c) 2 (d) 4
  26. 26.A drawer has lots of white, black, 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) 4 (c) 8 (d) 5
  27. 27.A drawer has lots of black, 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) 4 (b) 5 (c) 2 (d) 8
  28. 28.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) 21 (b) 22 (c) 29 (d) 28
  29. 29.A bag has 3 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) 4 (b) 11 (c) 9 (d) 2
  30. 30.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) 2 (b) 5 (c) 3 (d) 4
  31. 31.A bag has 7 red balls and 3 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 2 (b) 9 (c) 10 (d) 8
  32. 32.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
  33. 33.A bag has 4 yellow balls and 12 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 2 (b) 6 (c) 5 (d) 16
  34. 34.A bag has 11 white balls and 9 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 20 (b) 2 (c) 12 (d) 13
  35. 35.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) 8 (b) 2 (c) 5 (d) 4
  36. 36.A drawer has lots of blue, yellow, black, 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) 5 (b) 2 (c) 6 (d) 10
  37. 37.A drawer has lots of white, blue 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) 6
  38. 38.A bag has 7 red 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) 12 (b) 18 (c) 8 (d) 2
  39. 39.A drawer has lots of 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) 5 (b) 4 (c) 2 (d) 3
  40. 40.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

Answer key

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

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