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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 50 · 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, white, 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) 5 (b) 10 (c) 6 (d) 2
  2. 2.A bag has 8 red balls and 6 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 2 (b) 9 (c) 10 (d) 14
  3. 3.A drawer has lots of black, yellow, red, 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) 5 (b) 10 (c) 6 (d) 2
  4. 4.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) 8 (d) 15
  5. 5.A drawer has lots of white, red, yellow, 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) 2 (b) 6 (c) 5 (d) 10
  6. 6.A drawer has lots of 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) 3 (c) 2 (d) 6
  7. 7.A drawer has lots of blue, green, yellow, 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) 10 (b) 5 (c) 2 (d) 6
  8. 8.A bag has 9 black balls and 5 blue 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) 14 (d) 6
  9. 9.A drawer has lots of 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) 4 (b) 2 (c) 6 (d) 3
  10. 10.A bag has 9 blue balls and 3 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) 10 (c) 4 (d) 12
  11. 11.A bag has 10 red balls and 7 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) 17 (c) 2 (d) 11
  12. 12.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) 12 (b) 13 (c) 24 (d) 25
  13. 13.A drawer has lots of green, black, yellow, 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) 6 (b) 2 (c) 10 (d) 5
  14. 14.A drawer has lots of black, 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) 6 (b) 3 (c) 2 (d) 4
  15. 15.A bag has 4 blue balls and 3 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 6 (b) 5 (c) 7 (d) 2
  16. 16.A drawer has lots of blue, black, green, 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) 5 (b) 6 (c) 2 (d) 10
  17. 17.A bag has 5 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) 2 (b) 6 (c) 8 (d) 12
  18. 18.A drawer has lots of yellow, 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) 6 (b) 2 (c) 3 (d) 4
  19. 19.A bag has 6 green balls and 7 yellow 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) 13 (d) 8
  20. 20.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) 37 (c) 24 (d) 25
  21. 21.A bag has 12 green balls and 10 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 14 (b) 22 (c) 13 (d) 2
  22. 22.A bag has 9 blue balls and 3 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 10 (b) 11 (c) 12 (d) 2
  23. 23.A drawer has lots of green, 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) 8 (b) 4 (c) 5 (d) 2
  24. 24.A bag has 12 white balls and 8 green 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) 2 (c) 13 (d) 20
  25. 25.A bag has 10 white balls and 12 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 12 (b) 11 (c) 22 (d) 2
  26. 26.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) 2 (c) 8 (d) 5
  27. 27.A bag has 3 red balls and 3 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 2 (b) 5 (c) 6 (d) 4
  28. 28.A bag has 6 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) 7 (b) 9 (c) 4 (d) 2
  29. 29.A drawer has lots of green, 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) 2 (b) 6 (c) 3 (d) 4
  30. 30.A bag has 10 red balls and 3 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 2 (b) 13 (c) 12 (d) 11
  31. 31.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) 22 (c) 14 (d) 15
  32. 32.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) 36 (b) 49 (c) 37 (d) 48
  33. 33.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) 3 (b) 4 (c) 2 (d) 5
  34. 34.A drawer has lots of red, white, green, 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) 10 (b) 6 (c) 2 (d) 5
  35. 35.A drawer has lots of yellow, 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) 6 (b) 3 (c) 2 (d) 4
  36. 36.A drawer has lots of green, 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
  37. 37.A bag has 10 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) 2 (b) 13 (c) 22 (d) 11
  38. 38.A bag has 11 red balls and 8 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) 9 (c) 19 (d) 12
  39. 39.A bag has 9 blue balls and 4 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 10 (b) 11 (c) 13 (d) 2
  40. 40.A bag has 5 red balls and 12 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 17 (b) 6 (c) 2 (d) 7

Answer key

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

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