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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 423 · 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 8 green balls and 6 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 2 (b) 10 (c) 14 (d) 9
  2. 2.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) 13 (b) 25 (c) 12 (d) 24
  3. 3.A bag has 12 white balls and 9 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) 10 (c) 13 (d) 21
  4. 4.A drawer has lots of red, green, yellow, blue 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) 10 (b) 5 (c) 6 (d) 2
  5. 5.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
  6. 6.A drawer has lots of green, white, 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) 4 (b) 8 (c) 5 (d) 2
  7. 7.A bag has 6 blue balls and 8 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) 7 (c) 14 (d) 9
  8. 8.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) 22 (c) 21 (d) 14
  9. 9.A drawer has lots of 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) 6 (b) 4 (c) 3 (d) 2
  10. 10.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) 48 (b) 37 (c) 36 (d) 49
  11. 11.A drawer has lots of 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) 6 (b) 3 (c) 2 (d) 4
  12. 12.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) 14 (c) 8 (d) 7
  13. 13.A drawer has lots of blue, yellow, 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) 5 (b) 2 (c) 6 (d) 10
  14. 14.A bag has 5 black balls and 4 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) 2 (d) 5
  15. 15.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) 36 (d) 37
  16. 16.A drawer has lots of yellow, 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) 6 (b) 2 (c) 4 (d) 3
  17. 17.A bag has 5 red balls and 4 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) 6 (c) 2 (d) 9
  18. 18.A drawer has lots of yellow, 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) 3 (b) 6 (c) 4 (d) 2
  19. 19.A bag has 5 black 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) 11 (c) 6 (d) 15
  20. 20.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) 3 (b) 4 (c) 2 (d) 5
  21. 21.A drawer has lots of yellow, white, 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) 8 (c) 2 (d) 5
  22. 22.A drawer has lots of white, 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) 3 (c) 6 (d) 4
  23. 23.A bag has 3 white 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) 4 (c) 11 (d) 2
  24. 24.A bag has 9 green balls and 6 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 2 (b) 11 (c) 15 (d) 10
  25. 25.A bag has 10 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) 2 (c) 12 (d) 13
  26. 26.A drawer has lots of white, 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) 6 (b) 5 (c) 2 (d) 10
  27. 27.A drawer has lots of blue, white, green, 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) 5 (b) 2 (c) 6 (d) 10
  28. 28.A bag has 6 yellow balls and 7 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 7 (b) 13 (c) 8 (d) 2
  29. 29.A drawer has lots of black, green, red, 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) 5 (b) 6 (c) 2 (d) 10
  30. 30.A bag has 6 blue balls and 12 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 8 (b) 2 (c) 18 (d) 7
  31. 31.A bag has 12 blue 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) 2 (c) 7 (d) 18
  32. 32.A bag has 5 green 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) 6 (c) 2 (d) 7
  33. 33.A drawer has lots of blue, green, yellow, 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) 6 (b) 2 (c) 10 (d) 5
  34. 34.A drawer has lots of 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) 3 (c) 6 (d) 4
  35. 35.A bag has 12 black balls and 5 blue 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) 17 (c) 2 (d) 13
  36. 36.A drawer has lots of black, 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) 3 (b) 6 (c) 2 (d) 4
  37. 37.A drawer has lots of green, blue 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) 6 (d) 3
  38. 38.A drawer has lots of green, blue, yellow, 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) 10 (b) 5 (c) 6 (d) 2
  39. 39.A drawer has lots of blue 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
  40. 40.A drawer has lots of blue, 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) 3 (c) 6 (d) 2

Answer key

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

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