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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 320 · 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 7 green balls and 12 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 9 (b) 2 (c) 8 (d) 19
  2. 2.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) 24 (c) 25 (d) 13
  3. 3.A bag has 11 yellow balls and 4 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) 12 (d) 15
  4. 4.A bag has 4 yellow balls and 7 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 2 (b) 5 (c) 6 (d) 11
  5. 5.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
  6. 6.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
  7. 7.A drawer has lots of 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) 4 (c) 3 (d) 2
  8. 8.A drawer has lots of white, 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) 6 (c) 3 (d) 4
  9. 9.A bag has 8 white balls and 3 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 10 (b) 9 (c) 2 (d) 11
  10. 10.A drawer has lots of green, blue, 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) 8 (b) 4 (c) 5 (d) 2
  11. 11.A drawer has lots of 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) 2 (d) 3
  12. 12.A drawer has lots of black, 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) 8 (b) 5 (c) 4 (d) 2
  13. 13.A bag has 3 black balls and 8 blue balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 11 (b) 9 (c) 2 (d) 4
  14. 14.A drawer has lots of 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) 4 (b) 6 (c) 2 (d) 3
  15. 15.A bag has 6 red balls and 10 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) 11 (d) 16
  16. 16.A drawer has lots of red, yellow, 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) 2 (b) 5 (c) 10 (d) 6
  17. 17.A drawer has lots of 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) 6 (b) 4 (c) 3 (d) 2
  18. 18.A bag has 3 white balls and 4 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) 5
  19. 19.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) 8 (b) 4 (c) 2 (d) 5
  20. 20.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) 37 (b) 48 (c) 36 (d) 49
  21. 21.A bag has 5 yellow balls and 5 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 10 (b) 2 (c) 7 (d) 6
  22. 22.A bag has 5 red 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) 8 (c) 6 (d) 4
  23. 23.A drawer has lots of yellow, 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) 2 (b) 4 (c) 8 (d) 5
  24. 24.A bag has 9 yellow balls and 6 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 2 (b) 11 (c) 10 (d) 15
  25. 25.A bag has 11 black balls and 3 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 12 (b) 2 (c) 13 (d) 14
  26. 26.A bag has 11 black 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) 12 (b) 2 (c) 7 (d) 17
  27. 27.A bag has 9 yellow 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) 16 (c) 2 (d) 10
  28. 28.A bag has 9 green balls and 6 red balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 15 (b) 7 (c) 10 (d) 2
  29. 29.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) 3 (b) 2 (c) 5 (d) 4
  30. 30.A bag has 10 green balls and 3 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 11 (b) 12 (c) 13 (d) 2
  31. 31.A bag has 9 white balls and 11 green 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) 20 (c) 2 (d) 10
  32. 32.A bag has 9 yellow balls and 8 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) 9 (d) 17
  33. 33.A bag has 7 red balls and 8 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) 15
  34. 34.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) 3 (d) 4
  35. 35.A drawer has lots of yellow, 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) 2 (b) 4 (c) 3 (d) 6
  36. 36.A bag has 11 white balls and 12 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 12 (b) 2 (c) 23 (d) 13
  37. 37.A bag has 6 white 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) 2 (b) 10 (c) 5 (d) 7
  38. 38.A bag has 4 blue 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) 14 (b) 5 (c) 2 (d) 11
  39. 39.A bag has 10 yellow balls and 11 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 21 (b) 2 (c) 11 (d) 12
  40. 40.A drawer has lots of black, 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) 4 (c) 2 (d) 3

Answer key

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

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