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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 367 · 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 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) 5 (c) 4 (d) 3
  2. 2.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) 4 (b) 5 (c) 3 (d) 2
  3. 3.A drawer has lots of blue, yellow, white, 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) 10 (b) 5 (c) 6 (d) 2
  4. 4.A bag has 12 red balls and 3 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 14 (b) 13 (c) 15 (d) 2
  5. 5.A bag has 9 green balls and 9 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 2 (b) 18 (c) 10 (d) 11
  6. 6.A drawer has lots of yellow, white, red, blue 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) 10 (c) 5 (d) 2
  7. 7.A bag has 7 blue 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) 15 (b) 2 (c) 8 (d) 9
  8. 8.A bag has 4 blue balls and 5 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) 5 (c) 9 (d) 6
  9. 9.A drawer has lots of yellow, green, 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) 5 (b) 6 (c) 2 (d) 10
  10. 10.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) 4 (b) 5 (c) 2 (d) 3
  11. 11.A bag has 8 red balls and 8 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 16 (b) 9 (c) 2 (d) 10
  12. 12.A bag has 4 black balls and 7 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) 8 (c) 11 (d) 2
  13. 13.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) 22 (b) 21 (c) 29 (d) 28
  14. 14.A bag has 7 yellow 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) 8 (b) 9 (c) 2 (d) 15
  15. 15.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) 25 (d) 12
  16. 16.A drawer has lots of white, green, 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) 2 (b) 8 (c) 4 (d) 5
  17. 17.A bag has 6 red balls and 11 green 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) 7 (c) 12 (d) 2
  18. 18.A bag has 5 green balls and 3 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 8 (b) 2 (c) 7 (d) 6
  19. 19.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) 36 (c) 37 (d) 49
  20. 20.A bag has 8 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) 9 (b) 17 (c) 10 (d) 2
  21. 21.A bag has 11 green balls and 3 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 2 (b) 14 (c) 13 (d) 12
  22. 22.A drawer has lots of green, black, 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) 6 (b) 2 (c) 5 (d) 10
  23. 23.A bag has 3 black balls and 3 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 2 (b) 5 (c) 6 (d) 4
  24. 24.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) 7 (b) 2 (c) 11 (d) 16
  25. 25.A bag has 6 red balls and 12 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 2 (b) 18 (c) 7 (d) 8
  26. 26.A bag has 5 yellow balls and 6 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 2 (b) 7 (c) 11 (d) 6
  27. 27.A bag has 5 green balls and 4 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 7 (b) 2 (c) 6 (d) 9
  28. 28.A drawer has lots of blue, green, 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) 10 (b) 5 (c) 2 (d) 6
  29. 29.A bag has 5 white balls and 10 black 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) 15 (c) 6 (d) 2
  30. 30.A bag has 12 green balls and 10 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 22 (b) 13 (c) 2 (d) 14
  31. 31.A bag has 9 blue 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) 10 (b) 15 (c) 2 (d) 7
  32. 32.A drawer has lots of black, 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) 2 (b) 6 (c) 3 (d) 4
  33. 33.A bag has 5 green balls and 9 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 14 (b) 2 (c) 6 (d) 7
  34. 34.A bag has 8 white 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) 12 (b) 9 (c) 19 (d) 2
  35. 35.A bag has 3 yellow 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) 9 (c) 4 (d) 5
  36. 36.A bag has 9 white 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) 13 (b) 2 (c) 5 (d) 10
  37. 37.A bag has 3 yellow balls and 6 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 9 (b) 5 (c) 4 (d) 2
  38. 38.A drawer has lots of white, 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) 2 (b) 3 (c) 4 (d) 6
  39. 39.A bag has 6 green balls and 4 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 10 (b) 7 (c) 2 (d) 8
  40. 40.A bag has 3 yellow balls and 4 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) 7 (d) 2

Answer key

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

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