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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 696 · 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, yellow, blue, red 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) 10 (b) 2 (c) 6 (d) 5
  2. 2.A drawer has lots of yellow, white, red, 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) 10 (b) 5 (c) 2 (d) 6
  3. 3.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) 25 (d) 36
  4. 4.A bag has 9 green 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) 11 (c) 19 (d) 2
  5. 5.A bag has 7 yellow 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) 17 (b) 2 (c) 8 (d) 11
  6. 6.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) 4 (b) 6 (c) 2 (d) 3
  7. 7.A bag has 11 green balls and 4 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) 15 (c) 12 (d) 5
  8. 8.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) 2 (b) 4 (c) 5 (d) 3
  9. 9.A bag has 12 white 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) 5 (b) 13 (c) 2 (d) 16
  10. 10.A drawer has lots of 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) 2 (b) 5 (c) 4 (d) 8
  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 month? (a) 49 (b) 36 (c) 48 (d) 37
  12. 12.A bag has 3 blue 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) 8 (b) 4 (c) 2 (d) 10
  13. 13.A bag has 6 red balls and 10 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 7 (b) 16 (c) 8 (d) 2
  14. 14.A drawer has lots of white, 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
  15. 15.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
  16. 16.A bag has 7 white balls and 9 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 2 (b) 8 (c) 16 (d) 9
  17. 17.A drawer has lots of green, white, 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) 10 (b) 5 (c) 6 (d) 2
  18. 18.A bag has 8 white balls and 9 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 17 (b) 9 (c) 10 (d) 2
  19. 19.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) 7 (c) 8 (d) 14
  20. 20.A drawer has lots of 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) 6 (b) 2 (c) 4 (d) 3
  21. 21.A bag has 7 blue balls and 3 black 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) 4 (d) 8
  22. 22.A drawer has lots of 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) 5 (b) 4 (c) 2 (d) 3
  23. 23.A drawer has lots of green, 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) 6 (c) 4 (d) 2
  24. 24.A bag has 12 blue balls and 9 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 14 (b) 13 (c) 21 (d) 2
  25. 25.A bag has 4 red 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) 5 (b) 12 (c) 9 (d) 2
  26. 26.A bag has 6 yellow balls and 4 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 10 (b) 7 (c) 8 (d) 2
  27. 27.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) 5 (c) 3 (d) 4
  28. 28.A bag has 3 black 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) 11 (b) 4 (c) 9 (d) 2
  29. 29.A drawer has lots of yellow 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) 4 (c) 5 (d) 2
  30. 30.A bag has 3 red 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) 4 (b) 7 (c) 5 (d) 2
  31. 31.A drawer has lots of green, white, 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) 2 (b) 4 (c) 8 (d) 5
  32. 32.A bag has 7 black balls and 11 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 2 (b) 9 (c) 18 (d) 8
  33. 33.A drawer has lots of black, green, 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) 4 (b) 5 (c) 2 (d) 8
  34. 34.A bag has 4 white balls and 3 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) 7 (c) 4 (d) 5
  35. 35.A bag has 6 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) 7 (c) 2 (d) 13
  36. 36.A bag has 3 green balls and 9 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 2 (b) 4 (c) 5 (d) 12
  37. 37.A bag has 4 white 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) 2 (b) 5 (c) 11 (d) 8
  38. 38.A bag has 4 green balls and 4 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 2 (b) 5 (c) 8 (d) 6
  39. 39.A drawer has lots of yellow, white, 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) 2 (c) 8 (d) 5
  40. 40.A drawer has lots of blue, red, yellow, 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) 2 (b) 6 (c) 10 (d) 5

Answer key

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

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