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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 241 · 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 black, 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) 6 (c) 10 (d) 2
  2. 2.A drawer has lots of blue, white, 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) 4 (b) 2 (c) 5 (d) 8
  3. 3.A bag has 8 white balls and 9 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 2 (b) 10 (c) 9 (d) 17
  4. 4.A bag has 8 blue balls and 9 white 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) 2 (d) 10
  5. 5.A bag has 3 red balls and 7 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 2 (b) 10 (c) 4 (d) 5
  6. 6.A drawer has lots of 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) 3 (b) 2 (c) 5 (d) 4
  7. 7.A drawer has lots of white, yellow, green, 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) 5 (b) 6 (c) 10 (d) 2
  8. 8.A bag has 4 yellow balls and 11 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 2 (b) 5 (c) 15 (d) 6
  9. 9.A bag has 9 black balls and 6 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) 7 (c) 2 (d) 10
  10. 10.A drawer has lots of 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) 2 (b) 5 (c) 3 (d) 4
  11. 11.A bag has 3 green 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) 4 (c) 2 (d) 10
  12. 12.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) 25 (b) 37 (c) 24 (d) 36
  13. 13.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) 3 (c) 2 (d) 5
  14. 14.A drawer has lots of white, 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) 4 (b) 8 (c) 2 (d) 5
  15. 15.A drawer has lots of green, black, yellow, 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) 2 (b) 5 (c) 10 (d) 6
  16. 16.A drawer has lots of 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) 2 (b) 4 (c) 3 (d) 5
  17. 17.A bag has 3 red balls and 8 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) 4 (c) 11 (d) 9
  18. 18.A bag has 4 blue balls and 6 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 5 (b) 2 (c) 6 (d) 10
  19. 19.A drawer has lots of black, 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) 2 (c) 8 (d) 5
  20. 20.A drawer has lots of yellow, blue, 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) 4 (b) 5 (c) 8 (d) 2
  21. 21.A drawer has lots of 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) 3 (b) 2 (c) 4 (d) 5
  22. 22.A drawer has lots of 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) 2 (b) 4 (c) 3 (d) 5
  23. 23.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) 7 (d) 8
  24. 24.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) 3 (b) 4 (c) 6 (d) 2
  25. 25.A bag has 6 white 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) 9 (c) 7 (d) 4
  26. 26.A drawer has lots of black, 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) 4 (b) 3 (c) 6 (d) 2
  27. 27.A bag has 11 blue balls and 12 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) 12 (c) 23 (d) 13
  28. 28.A bag has 11 white balls and 9 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 12 (b) 20 (c) 13 (d) 2
  29. 29.A bag has 6 green 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) 7 (b) 2 (c) 5 (d) 10
  30. 30.A bag has 10 blue balls and 9 black balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 19 (b) 2 (c) 11 (d) 10
  31. 31.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) 21 (b) 28 (c) 29 (d) 22
  32. 32.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) 22 (b) 14 (c) 15 (d) 21
  33. 33.A bag has 9 blue balls and 8 white 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) 2 (c) 9 (d) 10
  34. 34.A drawer has lots of 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) 4 (b) 3 (c) 6 (d) 2
  35. 35.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) 8 (b) 2 (c) 15 (d) 9
  36. 36.A drawer has lots of black, 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) 4 (b) 2 (c) 8 (d) 5
  37. 37.A drawer has lots of 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) 3 (b) 6 (c) 4 (d) 2
  38. 38.A bag has 5 white balls and 9 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) 6 (c) 14 (d) 10
  39. 39.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) 12 (b) 24 (c) 13 (d) 25
  40. 40.A drawer has lots of white, blue, 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) 4 (b) 2 (c) 8 (d) 5

Answer key

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

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