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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 485 · 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 10 green balls and 4 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 11 (b) 12 (c) 2 (d) 14
  2. 2.A bag has 8 green balls and 3 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 11 (b) 10 (c) 9 (d) 2
  3. 3.A drawer has lots of 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) 6 (b) 2 (c) 4 (d) 3
  4. 4.A drawer has lots of red, 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) 6 (b) 2 (c) 3 (d) 4
  5. 5.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) 4 (b) 2 (c) 5 (d) 3
  6. 6.A drawer has lots of 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) 4 (c) 8 (d) 5
  7. 7.A drawer has lots of green, red, blue, 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) 6 (c) 5 (d) 10
  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 day of the week? (a) 28 (b) 21 (c) 29 (d) 22
  9. 9.A bag has 9 red balls and 4 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) 10 (c) 13 (d) 2
  10. 10.A bag has 7 blue balls and 3 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 2 (b) 8 (c) 9 (d) 10
  11. 11.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) 22 (b) 14 (c) 21 (d) 15
  12. 12.A bag has 10 green balls and 12 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 22 (b) 11 (c) 12 (d) 2
  13. 13.A drawer has lots of white, black, 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) 8 (b) 4 (c) 2 (d) 5
  14. 14.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) 8 (d) 14
  15. 15.A drawer has lots of blue, red, 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) 10 (b) 5 (c) 2 (d) 6
  16. 16.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) 5 (c) 2 (d) 3
  17. 17.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) 3 (b) 5 (c) 4 (d) 2
  18. 18.A drawer has lots of 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) 4 (b) 2 (c) 3 (d) 5
  19. 19.A bag has 3 blue balls and 7 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 2 (b) 4 (c) 5 (d) 10
  20. 20.A bag has 9 white balls and 12 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 11 (b) 21 (c) 10 (d) 2
  21. 21.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) 36 (c) 37 (d) 48
  22. 22.A bag has 4 blue 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) 13 (c) 10 (d) 5
  23. 23.A bag has 4 blue balls and 3 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) 5 (c) 2 (d) 7
  24. 24.A drawer has lots of green, red, 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) 5 (b) 8 (c) 4 (d) 2
  25. 25.A bag has 7 red 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) 8 (b) 2 (c) 15 (d) 9
  26. 26.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) 24 (c) 25 (d) 37
  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) 3 (b) 2 (c) 4 (d) 6
  28. 28.A bag has 3 green balls and 5 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 5 (b) 2 (c) 4 (d) 8
  29. 29.A drawer has lots of black, red, 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) 8 (b) 4 (c) 2 (d) 5
  30. 30.A bag has 3 blue balls and 5 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 4 (b) 5 (c) 8 (d) 2
  31. 31.A drawer has lots of 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) 4 (c) 5 (d) 3
  32. 32.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) 2 (b) 6 (c) 3 (d) 4
  33. 33.A bag has 6 red 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) 2 (b) 7 (c) 13 (d) 8
  34. 34.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) 2 (b) 4 (c) 5 (d) 3
  35. 35.A drawer has lots of white, yellow, 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) 2 (b) 6 (c) 10 (d) 5
  36. 36.A bag has 7 black balls and 7 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 8 (b) 14 (c) 9 (d) 2
  37. 37.A drawer has lots of green, red, black, 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) 10 (b) 5 (c) 2 (d) 6
  38. 38.A drawer has lots of 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) 5 (b) 2 (c) 3 (d) 4
  39. 39.A bag has 12 red balls and 6 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) 18
  40. 40.A bag has 5 black balls and 9 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 7 (b) 6 (c) 14 (d) 2

Answer key

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

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