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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 175 · 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 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) 5 (b) 4 (c) 3 (d) 2
  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 day of the week? (a) 14 (b) 22 (c) 15 (d) 21
  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) 7 (b) 15 (c) 14 (d) 8
  4. 4.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
  5. 5.A bag has 9 red balls and 8 white 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) 10 (d) 9
  6. 6.A bag has 11 yellow 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) 13 (b) 23 (c) 2 (d) 12
  7. 7.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) 28 (c) 29 (d) 22
  8. 8.A drawer has lots of black, 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) 2 (c) 6 (d) 3
  9. 9.A drawer has lots of black, white 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
  10. 10.A bag has 11 yellow balls and 8 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) 9 (c) 12 (d) 19
  11. 11.A bag has 8 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) 2 (b) 15 (c) 9 (d) 8
  12. 12.A bag has 3 blue balls and 9 yellow 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) 4 (c) 10 (d) 2
  13. 13.A bag has 3 green balls and 12 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 4 (b) 15 (c) 2 (d) 5
  14. 14.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
  15. 15.A drawer has lots of 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) 3
  16. 16.A bag has 8 yellow balls and 6 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 10 (b) 14 (c) 9 (d) 2
  17. 17.A bag has 5 blue balls and 6 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 11 (b) 6 (c) 2 (d) 7
  18. 18.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
  19. 19.A drawer has lots of black, yellow, 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) 8 (c) 4 (d) 5
  20. 20.A bag has 12 blue balls and 11 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 23 (b) 14 (c) 2 (d) 13
  21. 21.A bag has 12 red balls and 7 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) 19 (c) 8 (d) 13
  22. 22.A bag has 11 white balls and 12 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 2 (b) 23 (c) 13 (d) 12
  23. 23.A drawer has lots of white, 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) 2 (b) 4 (c) 3 (d) 6
  24. 24.A drawer has lots of red, yellow, 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) 5 (b) 6 (c) 10 (d) 2
  25. 25.A drawer has lots of black, 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) 4 (b) 3 (c) 2 (d) 6
  26. 26.A bag has 10 white balls and 9 green 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) 19 (d) 11
  27. 27.A drawer has lots of black, white 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) 6 (c) 4 (d) 3
  28. 28.A bag has 7 black balls and 8 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 8 (b) 2 (c) 9 (d) 15
  29. 29.A bag has 12 black balls and 10 yellow 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) 2 (c) 22 (d) 13
  30. 30.A drawer has lots of blue, black, green, 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) 6 (b) 5 (c) 2 (d) 10
  31. 31.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) 2 (c) 3 (d) 4
  32. 32.A bag has 12 red balls and 9 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 21 (b) 2 (c) 14 (d) 13
  33. 33.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) 13 (c) 25 (d) 12
  34. 34.A bag has 4 white 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) 5 (b) 7 (c) 4 (d) 2
  35. 35.A drawer has lots of black, red, 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) 4 (b) 2 (c) 5 (d) 8
  36. 36.A bag has 6 red balls and 7 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) 13 (c) 8 (d) 7
  37. 37.A bag has 9 blue balls and 11 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 11 (b) 2 (c) 10 (d) 20
  38. 38.A bag has 11 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) 11 (b) 21 (c) 2 (d) 12
  39. 39.A bag has 6 green 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) 9 (b) 7 (c) 2 (d) 4
  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) 5 (b) 4 (c) 2 (d) 3

Answer key

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

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