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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 687 · 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 four of them were born in the same day of the week? (a) 28 (b) 22 (c) 21 (d) 29
  2. 2.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) 24 (b) 36 (c) 25 (d) 37
  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 day of the week? (a) 14 (b) 15 (c) 7 (d) 8
  4. 4.A drawer has lots of yellow, 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) 6
  5. 5.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) 21 (b) 15 (c) 14 (d) 22
  6. 6.A bag has 3 blue balls and 3 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 5 (b) 6 (c) 2 (d) 4
  7. 7.A bag has 12 green balls and 5 blue 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) 17 (c) 13 (d) 2
  8. 8.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) 5 (b) 2 (c) 3 (d) 4
  9. 9.A drawer has lots of white, green, black, 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) 5 (b) 2 (c) 10 (d) 6
  10. 10.A drawer has lots of yellow, blue, black, green 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) 6 (c) 5 (d) 2
  11. 11.A bag has 10 red balls and 8 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 11 (b) 2 (c) 12 (d) 18
  12. 12.A drawer has lots of red, 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) 3 (b) 4 (c) 2 (d) 6
  13. 13.A bag has 7 red balls and 5 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) 6 (c) 12 (d) 8
  14. 14.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) 3 (c) 2 (d) 4
  15. 15.A drawer has lots of red, green, 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) 8 (b) 5 (c) 4 (d) 2
  16. 16.A bag has 11 blue 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) 13 (c) 15 (d) 12
  17. 17.A bag has 12 red balls and 3 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 13 (b) 2 (c) 15 (d) 14
  18. 18.A bag has 5 white balls and 10 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 15 (b) 7 (c) 2 (d) 6
  19. 19.A bag has 6 white balls and 11 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 8 (b) 7 (c) 17 (d) 2
  20. 20.A drawer has lots of 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) 2 (b) 3 (c) 6 (d) 4
  21. 21.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) 2 (b) 3 (c) 5 (d) 4
  22. 22.A bag has 4 blue balls and 4 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 6 (b) 5 (c) 8 (d) 2
  23. 23.A bag has 6 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) 7 (b) 8 (c) 2 (d) 13
  24. 24.A bag has 6 red 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) 7 (b) 9 (c) 4 (d) 2
  25. 25.A bag has 4 white 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) 2 (b) 7 (c) 5 (d) 4
  26. 26.A bag has 10 yellow balls and 8 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 11 (b) 2 (c) 18 (d) 12
  27. 27.A bag has 3 white balls and 5 blue 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) 6 (c) 4 (d) 2
  28. 28.A drawer has lots of 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) 3 (c) 2 (d) 5
  29. 29.A bag has 6 green balls and 3 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 9 (b) 7 (c) 2 (d) 8
  30. 30.A bag has 9 white balls and 3 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) 10 (c) 12 (d) 2
  31. 31.A drawer has lots of black, yellow, 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) 2 (b) 10 (c) 5 (d) 6
  32. 32.A bag has 3 green balls and 10 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) 11 (c) 2 (d) 13
  33. 33.A drawer has lots of green, 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) 8 (b) 4 (c) 5 (d) 2
  34. 34.A drawer has lots of blue, red, 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) 4 (b) 8 (c) 2 (d) 5
  35. 35.A drawer has lots of white, red, 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) 5 (c) 2 (d) 4
  36. 36.A bag has 10 black balls and 8 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 18 (b) 12 (c) 2 (d) 11
  37. 37.A drawer has lots of black, blue 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
  38. 38.A bag has 3 black balls and 10 blue 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) 13 (c) 2 (d) 11
  39. 39.A bag has 3 white balls and 11 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 5 (b) 4 (c) 2 (d) 14
  40. 40.A drawer has lots of yellow, green, blue, 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) 6 (d) 10

Answer key

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

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