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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 532 · 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, 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) 4 (b) 3 (c) 2 (d) 6
  2. 2.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) 2 (c) 3 (d) 5
  3. 3.A drawer has lots of red, green, 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) 4 (b) 5 (c) 8 (d) 2
  4. 4.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) 3 (b) 5 (c) 4 (d) 2
  5. 5.A drawer has lots of 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) 2 (b) 4 (c) 5 (d) 3
  6. 6.A bag has 7 black balls and 6 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 13 (b) 2 (c) 8 (d) 9
  7. 7.A drawer has lots of white, 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) 8 (c) 5 (d) 4
  8. 8.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) 14 (b) 7 (c) 15 (d) 8
  9. 9.A bag has 12 white balls and 11 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 23 (b) 14 (c) 13 (d) 2
  10. 10.A drawer has lots of 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) 3 (c) 5 (d) 4
  11. 11.A drawer has lots of 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) 4 (b) 2 (c) 3 (d) 5
  12. 12.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) 3 (b) 6 (c) 4 (d) 2
  13. 13.A drawer has lots of green, red, 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) 5 (b) 6 (c) 2 (d) 10
  14. 14.A bag has 10 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) 6 (b) 15 (c) 11 (d) 2
  15. 15.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) 13 (d) 25
  16. 16.A bag has 10 white balls and 9 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 19 (b) 2 (c) 12 (d) 11
  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) 4 (b) 2 (c) 6 (d) 3
  18. 18.A drawer has lots of red, 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) 10 (b) 2 (c) 5 (d) 6
  19. 19.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) 22 (d) 29
  20. 20.A drawer has lots of red, blue, 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) 4 (b) 5 (c) 8 (d) 2
  21. 21.A drawer has lots of 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) 3 (b) 6 (c) 2 (d) 4
  22. 22.A bag has 6 yellow 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) 17 (b) 12 (c) 7 (d) 2
  23. 23.A bag has 4 blue 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) 5 (c) 14 (d) 6
  24. 24.A bag has 3 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) 8 (b) 4 (c) 2 (d) 10
  25. 25.A bag has 10 blue balls and 11 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) 12 (c) 11 (d) 21
  26. 26.A drawer has lots of black, blue, red, 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) 10 (b) 6 (c) 5 (d) 2
  27. 27.A drawer has lots of 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) 3 (b) 6 (c) 2 (d) 4
  28. 28.A bag has 9 red balls and 6 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 2 (b) 10 (c) 11 (d) 15
  29. 29.A drawer has lots of red, 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) 3 (b) 2 (c) 6 (d) 4
  30. 30.A drawer has lots of blue, 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) 6 (c) 2 (d) 4
  31. 31.A bag has 12 red 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) 10 (b) 2 (c) 21 (d) 13
  32. 32.A bag has 4 white balls and 5 red 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) 5 (c) 2 (d) 6
  33. 33.A drawer has lots of yellow, blue, 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) 4 (d) 2
  34. 34.A drawer has lots of blue, white, 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) 8 (b) 5 (c) 2 (d) 4
  35. 35.A bag has 4 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) 12 (b) 15 (c) 2 (d) 5
  36. 36.A bag has 12 white 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) 5 (b) 16 (c) 2 (d) 13
  37. 37.A bag has 4 white balls and 4 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 8 (b) 6 (c) 5 (d) 2
  38. 38.A drawer has lots of yellow, blue, white 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) 8 (b) 4 (c) 2 (d) 5
  39. 39.A drawer has lots of green, blue, black 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) 8 (d) 2
  40. 40.A bag has 8 red balls and 9 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) 10 (c) 17 (d) 2

Answer key

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

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