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Pigeonhole Principle Worksheet for Maths Olympiad

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

Answer key

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

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