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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 86 · 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 white, 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) 4 (b) 5 (c) 8 (d) 2
  2. 2.A bag has 3 red balls and 7 white 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) 10 (d) 4
  3. 3.A bag has 3 black balls and 7 red 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) 10 (c) 4 (d) 8
  4. 4.A bag has 11 yellow balls and 12 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) 23 (c) 12 (d) 13
  5. 5.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
  6. 6.A bag has 7 red balls and 12 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 2 (b) 8 (c) 19 (d) 9
  7. 7.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) 25 (c) 24 (d) 36
  8. 8.A bag has 9 blue balls and 7 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) 16
  9. 9.A drawer has lots of white, 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) 6
  10. 10.A bag has 12 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) 7 (b) 13 (c) 18 (d) 2
  11. 11.A drawer has lots of blue, yellow, 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) 8 (b) 5 (c) 2 (d) 4
  12. 12.A bag has 7 yellow 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) 8 (d) 4
  13. 13.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) 28 (b) 21 (c) 29 (d) 22
  14. 14.A bag has 4 green balls and 10 red 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) 2 (c) 11 (d) 14
  15. 15.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) 4 (b) 3 (c) 2 (d) 6
  16. 16.A drawer has lots of 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) 2 (b) 4 (c) 5 (d) 3
  17. 17.A bag has 4 blue balls and 9 red 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) 10 (d) 13
  18. 18.A drawer has lots of green, red, 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) 8 (b) 2 (c) 4 (d) 5
  19. 19.A drawer has lots of 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) 5 (b) 3 (c) 4 (d) 2
  20. 20.A drawer has lots of red, green, 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) 8 (b) 5 (c) 4 (d) 2
  21. 21.A bag has 7 black 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) 2 (b) 4 (c) 10 (d) 8
  22. 22.A drawer has lots of white, red, 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) 5 (c) 6 (d) 10
  23. 23.A bag has 11 yellow balls and 12 blue balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 13 (b) 12 (c) 23 (d) 2
  24. 24.A bag has 6 white balls and 12 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 7 (b) 2 (c) 18 (d) 8
  25. 25.A bag has 6 red balls and 3 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) 7 (c) 4 (d) 2
  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) 8 (b) 14 (c) 15 (d) 7
  27. 27.A drawer has lots of red, 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) 2 (b) 6 (c) 3 (d) 4
  28. 28.A drawer has lots of red, white, blue, 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) 10 (b) 5 (c) 2 (d) 6
  29. 29.A bag has 10 yellow balls and 12 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 11 (b) 22 (c) 2 (d) 12
  30. 30.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) 5 (b) 4 (c) 2 (d) 3
  31. 31.A bag has 6 blue balls and 11 red 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) 12 (c) 17 (d) 2
  32. 32.A drawer has lots of red, green, 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) 8 (c) 5 (d) 2
  33. 33.A bag has 10 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) 11 (c) 19 (d) 2
  34. 34.A drawer has lots of black, blue, red, 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) 10 (b) 6 (c) 5 (d) 2
  35. 35.A bag has 7 green balls and 11 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) 18 (c) 2 (d) 12
  36. 36.A drawer has lots of yellow, 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) 3 (b) 6 (c) 2 (d) 4
  37. 37.A drawer has lots of yellow, red, green, 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) 10 (c) 5 (d) 2
  38. 38.A drawer has lots of blue, 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) 4 (b) 3 (c) 2 (d) 6
  39. 39.A bag has 8 yellow 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) 13 (b) 6 (c) 9 (d) 2
  40. 40.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) 12 (d) 13

Answer key

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

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