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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 122 · 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 4 green balls and 10 red 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) 2 (c) 5 (d) 14
  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) 24 (b) 13 (c) 12 (d) 25
  3. 3.A drawer has lots of white, 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) 4 (b) 8 (c) 5 (d) 2
  4. 4.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
  5. 5.A bag has 6 green 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) 2 (b) 9 (c) 4 (d) 7
  6. 6.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) 4 (c) 2 (d) 5
  7. 7.A drawer has lots of 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) 6 (b) 2 (c) 3 (d) 4
  8. 8.A bag has 7 blue balls and 11 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 2 (b) 8 (c) 9 (d) 18
  9. 9.A drawer has lots of blue, yellow, white, 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) 6 (d) 2
  10. 10.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) 37 (b) 24 (c) 36 (d) 25
  11. 11.A drawer has lots of 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) 4 (b) 8 (c) 5 (d) 2
  12. 12.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) 49 (d) 37
  13. 13.A drawer has lots of green, white, 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) 5 (c) 8 (d) 2
  14. 14.A bag has 4 black 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) 14 (b) 11 (c) 2 (d) 5
  15. 15.A drawer has lots of 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) 3 (c) 2 (d) 4
  16. 16.A drawer has lots of white, green, 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) 2 (b) 10 (c) 6 (d) 5
  17. 17.A bag has 4 red balls and 4 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 5 (b) 2 (c) 6 (d) 8
  18. 18.A bag has 9 black balls and 7 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 10 (b) 16 (c) 2 (d) 11
  19. 19.A bag has 9 green 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) 10 (b) 2 (c) 19 (d) 11
  20. 20.A drawer has lots of 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) 3 (b) 2 (c) 4 (d) 6
  21. 21.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) 14 (c) 8 (d) 15
  22. 22.A drawer has lots of 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) 5 (b) 4 (c) 2 (d) 3
  23. 23.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) 14 (d) 21
  24. 24.A bag has 8 red balls and 11 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 2 (b) 10 (c) 19 (d) 9
  25. 25.A bag has 5 blue balls and 12 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 2 (b) 7 (c) 6 (d) 17
  26. 26.A bag has 10 black balls and 3 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 12 (b) 11 (c) 2 (d) 13
  27. 27.A bag has 10 black balls and 11 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 11 (b) 12 (c) 2 (d) 21
  28. 28.A bag has 4 yellow balls and 6 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 6 (b) 2 (c) 5 (d) 10
  29. 29.A bag has 10 blue balls and 9 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 19 (b) 2 (c) 11 (d) 12
  30. 30.A bag has 11 black balls and 9 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 2 (b) 12 (c) 13 (d) 20
  31. 31.A bag has 5 green balls and 4 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 6 (b) 7 (c) 9 (d) 2
  32. 32.A drawer has lots of blue, 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) 2 (b) 5 (c) 4 (d) 8
  33. 33.A drawer has lots of blue, white, 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) 2 (b) 4 (c) 8 (d) 5
  34. 34.A drawer has lots of green, red, 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) 8 (b) 2 (c) 4 (d) 5
  35. 35.A bag has 5 blue 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) 13 (b) 9 (c) 6 (d) 2
  36. 36.A bag has 7 black balls and 3 red 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) 10 (d) 2
  37. 37.A drawer has lots of green, blue, 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) 6 (c) 5 (d) 10
  38. 38.A bag has 8 yellow balls and 3 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 11 (b) 2 (c) 9 (d) 10
  39. 39.A bag has 10 black balls and 8 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 12 (b) 2 (c) 11 (d) 18
  40. 40.A drawer has lots of 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) 4 (b) 5 (c) 2 (d) 3

Answer key

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

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