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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 88 · 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 three of them were born in the same month? (a) 24 (b) 25 (c) 37 (d) 36
  2. 2.A drawer has lots of green, 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) 3 (b) 2 (c) 4 (d) 6
  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) 24 (b) 25 (c) 13 (d) 12
  4. 4.A drawer has lots of red, white, yellow, 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) 6 (b) 2 (c) 10 (d) 5
  5. 5.A bag has 4 green balls and 5 yellow 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) 9 (c) 6 (d) 2
  6. 6.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) 5 (b) 4 (c) 7 (d) 2
  7. 7.A drawer has lots of yellow, green, 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) 5 (c) 2 (d) 8
  8. 8.A bag has 4 red balls and 3 yellow 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) 5 (c) 2 (d) 7
  9. 9.A bag has 8 white balls and 5 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 9 (b) 13 (c) 2 (d) 10
  10. 10.A bag has 3 blue 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) 4 (b) 9 (c) 7 (d) 2
  11. 11.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) 8 (b) 15 (c) 7 (d) 14
  12. 12.A bag has 4 green 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) 2 (b) 10 (c) 13 (d) 5
  13. 13.A drawer has lots of white, black, yellow, 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) 5 (b) 10 (c) 6 (d) 2
  14. 14.A drawer has lots of blue, black, 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) 5 (b) 8 (c) 4 (d) 2
  15. 15.A drawer has lots of red, 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) 2 (d) 5
  16. 16.A bag has 5 blue balls and 5 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 7 (b) 2 (c) 6 (d) 10
  17. 17.A bag has 10 yellow 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) 2 (b) 15 (c) 6 (d) 11
  18. 18.A bag has 4 black balls and 7 green 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) 5 (c) 8 (d) 2
  19. 19.A drawer has lots of green, white, black, 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) 2 (b) 6 (c) 5 (d) 10
  20. 20.A drawer has lots of white, yellow, black, 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) 2 (b) 10 (c) 5 (d) 6
  21. 21.A bag has 6 red balls and 7 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) 7 (c) 2 (d) 8
  22. 22.A bag has 4 black balls and 12 white 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) 16 (c) 2 (d) 13
  23. 23.A bag has 6 green balls and 4 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 2 (b) 10 (c) 7 (d) 8
  24. 24.A bag has 8 blue balls and 9 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 2 (b) 9 (c) 17 (d) 10
  25. 25.A bag has 9 white balls and 11 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 11 (b) 2 (c) 20 (d) 10
  26. 26.A bag has 12 yellow balls and 8 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 2 (b) 14 (c) 20 (d) 13
  27. 27.A drawer has lots of white, red, 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) 5 (b) 2 (c) 6 (d) 10
  28. 28.A bag has 7 blue balls and 10 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 9 (b) 17 (c) 8 (d) 2
  29. 29.A bag has 3 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) 4 (c) 8 (d) 6
  30. 30.A drawer has lots of black, red, blue, 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) 2 (c) 5 (d) 6
  31. 31.A drawer has lots of white, blue, red, yellow 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) 10 (c) 6 (d) 5
  32. 32.A bag has 5 yellow balls and 9 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 2 (b) 14 (c) 6 (d) 7
  33. 33.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) 49 (d) 36
  34. 34.A bag has 3 white balls and 9 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) 12 (c) 4 (d) 10
  35. 35.A drawer has lots of 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) 4 (b) 3 (c) 6 (d) 2
  36. 36.A bag has 7 white balls and 5 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 12 (b) 9 (c) 8 (d) 2
  37. 37.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) 3 (b) 5 (c) 2 (d) 4
  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 at least one ball of each colour? (a) 8 (b) 18 (c) 2 (d) 12
  39. 39.A bag has 4 green balls and 10 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 2 (b) 6 (c) 5 (d) 14
  40. 40.A bag has 10 blue balls and 4 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 12 (b) 2 (c) 14 (d) 11

Answer key

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

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