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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 989 · 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.

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  1. 1.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
  2. 2.A drawer has lots of blue, 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) 2 (b) 5 (c) 8 (d) 4
  3. 3.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) 21 (c) 14 (d) 22
  4. 4.A bag has 10 blue balls and 6 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 12 (b) 16 (c) 11 (d) 2
  5. 5.A bag has 5 black 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) 14 (d) 6
  6. 6.A bag has 8 green balls and 9 red balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 17 (b) 2 (c) 9 (d) 10
  7. 7.A bag has 10 green balls and 7 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 2 (b) 12 (c) 11 (d) 17
  8. 8.A bag has 4 yellow 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) 5 (b) 14 (c) 2 (d) 11
  9. 9.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) 36 (b) 25 (c) 37 (d) 24
  10. 10.A bag has 12 black balls and 10 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 13 (b) 14 (c) 2 (d) 22
  11. 11.A bag has 8 yellow 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) 9 (b) 4 (c) 11 (d) 2
  12. 12.A bag has 12 red balls and 7 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 2 (b) 19 (c) 14 (d) 13
  13. 13.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) 29 (b) 28 (c) 21 (d) 22
  14. 14.A drawer has lots of 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) 4 (b) 3 (c) 6 (d) 2
  15. 15.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) 49 (b) 48 (c) 37 (d) 36
  16. 16.A drawer has lots of green, white, 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) 5 (b) 2 (c) 8 (d) 4
  17. 17.A drawer has lots of green, 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) 2 (b) 3 (c) 4 (d) 6
  18. 18.A drawer has lots of 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) 5 (c) 3 (d) 2
  19. 19.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) 3 (b) 5 (c) 2 (d) 4
  20. 20.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) 10 (b) 2 (c) 8 (d) 4
  21. 21.A drawer has lots of yellow, red, blue, 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) 10 (b) 2 (c) 6 (d) 5
  22. 22.A drawer has lots of red, black, 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) 5 (b) 10 (c) 6 (d) 2
  23. 23.A bag has 5 yellow balls and 7 white 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) 12 (c) 2 (d) 8
  24. 24.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) 5 (b) 2 (c) 10 (d) 6
  25. 25.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) 2 (b) 5 (c) 4 (d) 3
  26. 26.A bag has 7 green balls and 9 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 16 (b) 9 (c) 8 (d) 2
  27. 27.A bag has 4 blue 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) 5 (c) 11 (d) 2
  28. 28.A drawer has lots of red, green, 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) 5 (b) 2 (c) 4 (d) 8
  29. 29.A bag has 7 blue balls and 6 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 13 (b) 2 (c) 8 (d) 9
  30. 30.A drawer has lots of red, 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) 3 (c) 6 (d) 4
  31. 31.A bag has 3 yellow balls and 11 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 2 (b) 14 (c) 4 (d) 5
  32. 32.A bag has 8 white balls and 12 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 10 (b) 2 (c) 20 (d) 9
  33. 33.A drawer has lots of white, green, 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) 2 (b) 4 (c) 5 (d) 8
  34. 34.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) 15 (b) 7 (c) 8 (d) 14
  35. 35.A bag has 6 green balls and 11 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 7 (b) 17 (c) 2 (d) 8
  36. 36.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) 6 (b) 2 (c) 5 (d) 10
  37. 37.A bag has 7 green 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) 8 (b) 2 (c) 15 (d) 9
  38. 38.A bag has 10 red 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) 11 (b) 14 (c) 5 (d) 2
  39. 39.A drawer has lots of red, black, 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) 8 (b) 4 (c) 5 (d) 2
  40. 40.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) 2 (b) 6 (c) 4 (d) 3

Answer key

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

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