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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 187 · 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, 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) 4 (b) 2 (c) 8 (d) 5
  2. 2.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) 21 (d) 15
  3. 3.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) 28 (c) 22 (d) 21
  4. 4.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) 24 (c) 37 (d) 25
  5. 5.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) 37 (d) 48
  6. 6.A drawer has lots of blue, white 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) 3 (c) 2 (d) 6
  7. 7.A bag has 8 yellow 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) 17 (b) 9 (c) 10 (d) 2
  8. 8.A bag has 5 white balls and 10 green 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) 6 (c) 11 (d) 2
  9. 9.A drawer has lots of red, green, 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) 4 (b) 2 (c) 8 (d) 5
  10. 10.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) 3 (b) 2 (c) 4 (d) 5
  11. 11.A bag has 11 red 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) 11 (b) 12 (c) 21 (d) 2
  12. 12.A bag has 3 yellow 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) 4 (b) 2 (c) 12 (d) 10
  13. 13.A drawer has lots of green, red, black, 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) 2 (b) 10 (c) 6 (d) 5
  14. 14.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) 13 (b) 24 (c) 25 (d) 12
  15. 15.A bag has 3 white 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) 12 (c) 4 (d) 2
  16. 16.A bag has 5 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) 2 (b) 6 (c) 9 (d) 13
  17. 17.A bag has 6 blue 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) 15 (d) 7
  18. 18.A bag has 6 blue balls and 3 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 7 (b) 9 (c) 8 (d) 2
  19. 19.A bag has 5 green balls and 6 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 11 (b) 2 (c) 6 (d) 7
  20. 20.A bag has 10 white balls and 8 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 12 (b) 2 (c) 11 (d) 18
  21. 21.A bag has 10 red 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) 11 (b) 4 (c) 13 (d) 2
  22. 22.A bag has 6 blue 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) 7 (d) 8
  23. 23.A drawer has lots of white, 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) 2 (b) 6 (c) 3 (d) 4
  24. 24.A drawer has lots of blue, black, 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) 8 (c) 4 (d) 5
  25. 25.A drawer has lots of yellow, 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) 4 (c) 3 (d) 6
  26. 26.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) 8 (c) 14 (d) 15
  27. 27.A drawer has lots of 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) 4 (b) 2 (c) 3 (d) 5
  28. 28.A drawer has lots of red, green, black, 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) 2 (b) 10 (c) 5 (d) 6
  29. 29.A bag has 4 white balls and 4 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 8 (b) 2 (c) 6 (d) 5
  30. 30.A bag has 10 blue balls and 3 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 13 (b) 11 (c) 12 (d) 2
  31. 31.A bag has 7 black balls and 9 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 8 (b) 16 (c) 9 (d) 2
  32. 32.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) 4 (b) 5 (c) 3 (d) 2
  33. 33.A bag has 5 red balls and 4 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 2 (b) 7 (c) 9 (d) 6
  34. 34.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) 3 (b) 4 (c) 2 (d) 6
  35. 35.A drawer has lots of yellow, black, red, 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) 10 (b) 2 (c) 6 (d) 5
  36. 36.A drawer has lots of blue, 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) 3 (c) 6 (d) 4
  37. 37.A bag has 7 yellow balls and 5 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 9 (b) 8 (c) 12 (d) 2
  38. 38.A drawer has lots of blue, red, black, 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) 10 (b) 6 (c) 2 (d) 5
  39. 39.A drawer has lots of white, red, 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) 2 (b) 4 (c) 8 (d) 5
  40. 40.A bag has 11 green balls and 4 red 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) 2 (c) 15 (d) 5

Answer key

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

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