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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 51 · 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 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) 4 (c) 5 (d) 2
  2. 2.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) 2 (b) 4 (c) 3 (d) 5
  3. 3.A bag has 3 yellow 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) 4 (c) 11 (d) 5
  4. 4.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) 25 (b) 24 (c) 13 (d) 12
  5. 5.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
  6. 6.A drawer has lots of white, 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) 4 (b) 2 (c) 6 (d) 3
  7. 7.A bag has 4 red balls and 9 green 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) 13 (d) 5
  8. 8.A bag has 3 black balls and 11 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 4 (b) 2 (c) 5 (d) 14
  9. 9.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) 4 (c) 3 (d) 6
  10. 10.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) 2 (b) 6 (c) 3 (d) 4
  11. 11.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) 29 (b) 21 (c) 28 (d) 22
  12. 12.A bag has 5 red balls and 12 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) 17
  13. 13.A bag has 5 green 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) 4 (c) 6 (d) 8
  14. 14.A bag has 8 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) 6 (b) 2 (c) 9 (d) 13
  15. 15.A bag has 6 black balls and 10 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 2 (b) 7 (c) 8 (d) 16
  16. 16.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) 25 (c) 37 (d) 24
  17. 17.A drawer has lots of 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) 3 (b) 2 (c) 5 (d) 4
  18. 18.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) 14 (b) 7 (c) 15 (d) 8
  19. 19.A bag has 8 white balls and 5 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 2 (b) 10 (c) 13 (d) 9
  20. 20.A bag has 4 white 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) 5 (b) 9 (c) 6 (d) 2
  21. 21.A bag has 3 black balls and 6 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 9 (b) 5 (c) 4 (d) 2
  22. 22.A bag has 12 blue balls and 3 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 13 (b) 14 (c) 2 (d) 15
  23. 23.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) 36 (b) 49 (c) 48 (d) 37
  24. 24.A bag has 11 yellow 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) 23 (b) 13 (c) 2 (d) 12
  25. 25.A bag has 4 white 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) 2 (b) 9 (c) 6 (d) 5
  26. 26.A bag has 9 white balls and 10 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 10 (b) 2 (c) 19 (d) 11
  27. 27.A drawer has lots of white, 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) 2 (b) 6 (c) 3 (d) 4
  28. 28.A drawer has lots of yellow, red, 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) 5 (b) 10 (c) 2 (d) 6
  29. 29.A bag has 8 red balls and 6 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 10 (b) 14 (c) 9 (d) 2
  30. 30.A drawer has lots of blue, 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) 3 (b) 4 (c) 6 (d) 2
  31. 31.A drawer has lots of 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) 2 (b) 3 (c) 4 (d) 5
  32. 32.A bag has 9 green balls and 3 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 2 (b) 12 (c) 11 (d) 10
  33. 33.A bag has 10 white balls and 6 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 12 (b) 2 (c) 11 (d) 16
  34. 34.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) 14 (b) 10 (c) 11 (d) 2
  35. 35.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) 9 (b) 2 (c) 6 (d) 5
  36. 36.A drawer has lots of red, blue, yellow, black 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) 2 (b) 5 (c) 10 (d) 6
  37. 37.A drawer has lots of blue, yellow, 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) 5 (b) 10 (c) 2 (d) 6
  38. 38.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
  39. 39.A bag has 3 black balls and 5 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 8 (b) 5 (c) 4 (d) 2
  40. 40.A bag has 5 green 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) 11 (b) 15 (c) 6 (d) 2

Answer key

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

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