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Pigeonhole Principle Worksheet for Maths Olympiad

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

Answer key

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

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