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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 236 · 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 green, blue, 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) 8 (c) 4 (d) 2
  2. 2.A bag has 12 black 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) 13 (c) 16 (d) 2
  3. 3.A bag has 3 blue 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) 2 (b) 4 (c) 7 (d) 5
  4. 4.A bag has 7 yellow balls and 6 white balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 7 (b) 13 (c) 2 (d) 8
  5. 5.A drawer has lots of white, red, black, 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) 10 (c) 5 (d) 6
  6. 6.A bag has 10 green balls and 12 red balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 22 (b) 11 (c) 2 (d) 13
  7. 7.A bag has 5 red 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) 8 (b) 4 (c) 2 (d) 6
  8. 8.A bag has 12 green balls and 3 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 15 (b) 14 (c) 13 (d) 2
  9. 9.A drawer has lots of blue, yellow, 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) 5 (d) 8
  10. 10.A bag has 4 red 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) 5 (b) 6 (c) 2 (d) 9
  11. 11.A bag has 6 green balls and 8 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 2 (b) 8 (c) 7 (d) 14
  12. 12.A bag has 11 blue balls and 9 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 20 (b) 13 (c) 2 (d) 12
  13. 13.A bag has 3 black balls and 12 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) 15 (c) 13 (d) 4
  14. 14.A bag has 12 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) 17 (b) 6 (c) 13 (d) 2
  15. 15.A drawer has lots of 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) 5 (d) 3
  16. 16.A drawer has lots of 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) 3 (c) 4 (d) 5
  17. 17.A bag has 9 red balls and 9 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 18 (b) 2 (c) 10 (d) 11
  18. 18.A bag has 4 blue 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) 2 (b) 7 (c) 5 (d) 4
  19. 19.A drawer has lots of black, white, 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) 10 (b) 2 (c) 6 (d) 5
  20. 20.A bag has 12 white 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) 8 (b) 19 (c) 2 (d) 13
  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 day of the week? (a) 28 (b) 29 (c) 21 (d) 22
  22. 22.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) 37 (c) 48 (d) 36
  23. 23.A bag has 8 green balls and 10 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 2 (b) 10 (c) 18 (d) 9
  24. 24.A drawer has lots of 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) 4 (b) 2 (c) 5 (d) 3
  25. 25.A bag has 4 red balls and 7 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) 8 (c) 2 (d) 5
  26. 26.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) 24 (c) 12 (d) 25
  27. 27.A bag has 4 black balls and 5 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) 6 (c) 2 (d) 5
  28. 28.A drawer has lots of 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) 3 (b) 4 (c) 2 (d) 6
  29. 29.A drawer has lots of blue, 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) 3 (c) 4 (d) 6
  30. 30.A bag has 4 blue balls and 11 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 15 (b) 6 (c) 2 (d) 5
  31. 31.A bag has 3 yellow balls and 11 blue 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) 14 (c) 2 (d) 12
  32. 32.A drawer has lots of white, blue, black, 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) 10 (d) 6
  33. 33.A bag has 9 red balls and 5 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 10 (b) 14 (c) 11 (d) 2
  34. 34.A bag has 9 blue 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) 2 (b) 10 (c) 21 (d) 13
  35. 35.A drawer has lots of red, 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) 4 (b) 3 (c) 6 (d) 2
  36. 36.A drawer has lots of yellow, 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) 6 (b) 3 (c) 2 (d) 4
  37. 37.A bag has 10 red balls and 6 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) 7 (c) 11 (d) 16
  38. 38.A bag has 10 black 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) 9 (b) 18 (c) 11 (d) 2
  39. 39.A bag has 6 red 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) 7 (b) 11 (c) 2 (d) 16
  40. 40.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) 5 (b) 2 (c) 8 (d) 4

Answer key

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

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