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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 139 · 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 bag has 5 blue balls and 12 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 17 (b) 2 (c) 7 (d) 6
  2. 2.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) 22 (c) 28 (d) 29
  3. 3.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) 4 (b) 3 (c) 2 (d) 5
  4. 4.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
  5. 5.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) 12 (b) 13 (c) 25 (d) 24
  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) 36 (b) 37 (c) 24 (d) 25
  7. 7.A drawer has lots of red, blue, 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) 5 (b) 2 (c) 10 (d) 6
  8. 8.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) 49 (c) 36 (d) 48
  9. 9.A drawer has lots of white, red, yellow, 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) 10 (c) 2 (d) 6
  10. 10.A bag has 3 red 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) 13 (b) 2 (c) 11 (d) 4
  11. 11.A drawer has lots of 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) 3 (b) 2 (c) 6 (d) 4
  12. 12.A bag has 9 yellow balls and 3 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) 4 (c) 10 (d) 12
  13. 13.A bag has 4 red balls and 12 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 5 (b) 6 (c) 16 (d) 2
  14. 14.A drawer has lots of red, 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) 3 (b) 2 (c) 6 (d) 4
  15. 15.A bag has 6 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) 5 (b) 7 (c) 10 (d) 2
  16. 16.A bag has 8 green balls and 12 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 9 (b) 2 (c) 20 (d) 10
  17. 17.A bag has 6 blue balls and 12 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 18 (b) 2 (c) 7 (d) 8
  18. 18.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) 5 (b) 4 (c) 2 (d) 3
  19. 19.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) 4 (b) 2 (c) 3 (d) 5
  20. 20.A drawer has lots of black, red, 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) 10 (b) 2 (c) 5 (d) 6
  21. 21.A bag has 7 black 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) 15 (d) 8
  22. 22.A bag has 9 white balls and 6 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 2 (b) 11 (c) 10 (d) 15
  23. 23.A bag has 5 white balls and 8 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) 6 (c) 13 (d) 2
  24. 24.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
  25. 25.A bag has 10 white balls and 7 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 17 (b) 12 (c) 11 (d) 2
  26. 26.A bag has 10 red 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) 11 (c) 14 (d) 12
  27. 27.A drawer has lots of 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) 4 (b) 3 (c) 6 (d) 2
  28. 28.A drawer has lots of blue, yellow, 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) 2 (d) 8
  29. 29.A bag has 8 blue balls and 11 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 19 (b) 9 (c) 2 (d) 10
  30. 30.A drawer has lots of red, 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) 3 (b) 2 (c) 4 (d) 6
  31. 31.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) 3 (b) 2 (c) 6 (d) 4
  32. 32.A bag has 4 green balls and 12 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 5 (b) 2 (c) 6 (d) 16
  33. 33.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) 2 (b) 3 (c) 4 (d) 6
  34. 34.A bag has 3 blue balls and 10 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) 11 (d) 13
  35. 35.A bag has 6 black balls and 10 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 2 (b) 8 (c) 16 (d) 7
  36. 36.A drawer has lots of red, yellow, 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) 5 (c) 4 (d) 8
  37. 37.A bag has 6 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) 6 (b) 11 (c) 2 (d) 7
  38. 38.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
  39. 39.A drawer has lots of green, 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) 2 (b) 6 (c) 4 (d) 3
  40. 40.A bag has 7 black balls and 3 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 10 (b) 8 (c) 2 (d) 9

Answer key

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

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