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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 464 · 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 bag has 10 black balls and 7 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 2 (b) 12 (c) 17 (d) 11
  2. 2.A drawer has lots of blue, 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) 3 (c) 4 (d) 6
  3. 3.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) 4 (b) 3 (c) 5 (d) 2
  4. 4.A drawer has lots of yellow, 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) 5 (b) 6 (c) 2 (d) 10
  5. 5.A drawer has lots of white, 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) 2 (c) 3 (d) 6
  6. 6.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
  7. 7.A bag has 5 red balls and 8 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 13 (b) 7 (c) 2 (d) 6
  8. 8.A bag has 6 white 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) 2 (b) 9 (c) 7 (d) 4
  9. 9.A bag has 10 green balls and 4 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 11 (b) 2 (c) 12 (d) 14
  10. 10.A bag has 11 yellow 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) 2 (d) 20
  11. 11.A drawer has lots of blue, black, green, 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) 10 (b) 5 (c) 6 (d) 2
  12. 12.A drawer has lots of red, black, 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) 5 (b) 4 (c) 8 (d) 2
  13. 13.A drawer has lots of green, blue, 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) 8 (b) 2 (c) 4 (d) 5
  14. 14.A bag has 11 red balls and 3 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 13 (b) 2 (c) 14 (d) 12
  15. 15.A drawer has lots of white, red, 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) 5 (b) 10 (c) 6 (d) 2
  16. 16.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) 37 (c) 48 (d) 36
  17. 17.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) 25 (d) 24
  18. 18.A bag has 12 blue balls and 3 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 15 (b) 13 (c) 2 (d) 14
  19. 19.A bag has 6 blue 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) 6 (b) 11 (c) 7 (d) 2
  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 day of the week? (a) 14 (b) 22 (c) 21 (d) 15
  21. 21.A bag has 12 blue balls and 4 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 13 (b) 2 (c) 16 (d) 14
  22. 22.A bag has 4 white balls and 10 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 14 (b) 2 (c) 6 (d) 5
  23. 23.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) 21 (b) 28 (c) 29 (d) 22
  24. 24.A drawer has lots of black, green, white, 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) 6 (b) 2 (c) 5 (d) 10
  25. 25.A bag has 3 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) 4 (b) 8 (c) 2 (d) 6
  26. 26.A bag has 10 yellow balls and 4 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 14 (b) 12 (c) 11 (d) 2
  27. 27.A drawer has lots of blue, green, 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) 5 (b) 6 (c) 10 (d) 2
  28. 28.A drawer has lots of 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) 2 (b) 5 (c) 4 (d) 3
  29. 29.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
  30. 30.A bag has 3 black balls and 5 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 2 (b) 4 (c) 5 (d) 8
  31. 31.A drawer has lots of 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) 2 (c) 5 (d) 3
  32. 32.A bag has 7 red balls and 11 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 9 (b) 8 (c) 18 (d) 2
  33. 33.A drawer has lots of green, yellow, red, 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) 6 (b) 10 (c) 2 (d) 5
  34. 34.A bag has 6 black balls and 3 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 7 (b) 9 (c) 8 (d) 2
  35. 35.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) 6 (b) 4 (c) 2 (d) 3
  36. 36.A drawer has lots of red, white, 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) 8 (b) 4 (c) 2 (d) 5
  37. 37.A drawer has lots of 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) 4 (b) 3 (c) 2 (d) 6
  38. 38.A bag has 6 yellow balls and 5 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 7 (b) 11 (c) 2 (d) 8
  39. 39.A drawer has lots of green, blue, white, 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) 5 (b) 10 (c) 2 (d) 6
  40. 40.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) 4 (c) 6 (d) 2

Answer key

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

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