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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 382 · 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 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) 6 (b) 3 (c) 2 (d) 4
  2. 2.A bag has 12 green balls and 9 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 13 (b) 21 (c) 14 (d) 2
  3. 3.A drawer has lots of 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) 5 (b) 3 (c) 2 (d) 4
  4. 4.A drawer has lots of 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) 5 (c) 3 (d) 2
  5. 5.A bag has 5 black balls and 4 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 7 (b) 2 (c) 6 (d) 9
  6. 6.A bag has 4 white balls and 12 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 6 (b) 16 (c) 5 (d) 2
  7. 7.A bag has 3 red 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) 4 (c) 5 (d) 2
  8. 8.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) 5 (b) 2 (c) 4 (d) 3
  9. 9.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) 2 (b) 4 (c) 5 (d) 3
  10. 10.A bag has 12 green balls and 13 black balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 25 (b) 14 (c) 13 (d) 2
  11. 11.A bag has 5 red 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) 17 (c) 6 (d) 2
  12. 12.A drawer has lots of 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) 3 (b) 2 (c) 4 (d) 5
  13. 13.A drawer has lots of black, green, 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) 8 (b) 5 (c) 4 (d) 2
  14. 14.A drawer has lots of red, blue, 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) 2 (b) 6 (c) 10 (d) 5
  15. 15.A bag has 10 green balls and 12 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 12 (b) 11 (c) 22 (d) 2
  16. 16.A drawer has lots of white, black, 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) 5 (c) 4 (d) 8
  17. 17.A drawer has lots of white, 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) 6 (b) 5 (c) 10 (d) 2
  18. 18.A bag has 11 white 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) 12 (b) 2 (c) 23 (d) 13
  19. 19.A bag has 12 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) 2 (b) 7 (c) 18 (d) 13
  20. 20.A bag has 6 white balls and 3 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 2 (b) 9 (c) 8 (d) 7
  21. 21.A bag has 8 green balls and 10 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 10 (b) 9 (c) 18 (d) 2
  22. 22.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) 22 (c) 21 (d) 29
  23. 23.A drawer has lots of yellow, blue, 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) 5 (b) 2 (c) 6 (d) 10
  24. 24.A drawer has lots of white, blue, red, 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) 5 (b) 6 (c) 2 (d) 10
  25. 25.A drawer has lots of red, 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) 3 (c) 4 (d) 6
  26. 26.A drawer has lots of white, yellow, black, 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) 5 (b) 2 (c) 6 (d) 10
  27. 27.A bag has 3 green balls and 4 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 2 (b) 7 (c) 5 (d) 4
  28. 28.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) 15 (d) 14
  29. 29.A drawer has lots of yellow, red, 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) 5 (b) 8 (c) 4 (d) 2
  30. 30.A bag has 3 red balls and 6 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) 7 (c) 2 (d) 9
  31. 31.A bag has 4 yellow balls and 10 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 2 (b) 14 (c) 5 (d) 6
  32. 32.A drawer has lots of black, 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) 5 (b) 2 (c) 8 (d) 4
  33. 33.A bag has 5 red balls and 10 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) 11 (c) 6 (d) 15
  34. 34.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) 36 (d) 25
  35. 35.A drawer has lots of green, 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
  36. 36.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) 12 (c) 25 (d) 13
  37. 37.A drawer has lots of green, 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) 8 (b) 2 (c) 5 (d) 4
  38. 38.A drawer has lots of 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) 5 (c) 4 (d) 3
  39. 39.A bag has 6 green balls and 6 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 7 (b) 12 (c) 8 (d) 2
  40. 40.A bag has 5 red balls and 7 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 12 (b) 6 (c) 2 (d) 7

Answer key

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

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