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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 376 · 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 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) 5 (c) 3 (d) 4
  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) 25 (b) 24 (c) 13 (d) 12
  3. 3.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) 49 (d) 36
  4. 4.A bag has 12 blue 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) 18 (b) 2 (c) 13 (d) 7
  5. 5.A bag has 6 white balls and 5 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 7 (b) 2 (c) 8 (d) 11
  6. 6.A bag has 12 black balls and 9 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 13 (b) 14 (c) 21 (d) 2
  7. 7.A bag has 9 white balls and 4 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 10 (b) 2 (c) 11 (d) 13
  8. 8.A drawer has lots of red, white, 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
  9. 9.A drawer has lots of green, white, blue, 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) 6 (b) 5 (c) 2 (d) 10
  10. 10.A bag has 9 yellow balls and 9 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 10 (b) 18 (c) 2 (d) 11
  11. 11.A drawer has lots of 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) 6 (b) 2 (c) 4 (d) 3
  12. 12.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) 3 (b) 5 (c) 2 (d) 4
  13. 13.A drawer has lots of black, 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) 4 (b) 2 (c) 8 (d) 5
  14. 14.A drawer has lots of red, yellow, 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) 4 (b) 2 (c) 5 (d) 8
  15. 15.A drawer has lots of blue, 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) 3 (b) 4 (c) 6 (d) 2
  16. 16.A bag has 3 green balls and 7 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 4 (b) 5 (c) 2 (d) 10
  17. 17.A drawer has lots of red, yellow, black, 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) 5 (b) 6 (c) 10 (d) 2
  18. 18.A drawer has lots of blue, 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) 5 (b) 8 (c) 4 (d) 2
  19. 19.A drawer has lots of yellow, red, black, 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) 6 (b) 5 (c) 10 (d) 2
  20. 20.A bag has 9 red balls and 3 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 10 (b) 2 (c) 12 (d) 11
  21. 21.A bag has 5 yellow balls and 11 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 6 (b) 2 (c) 7 (d) 16
  22. 22.A bag has 12 yellow balls and 9 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 13 (b) 14 (c) 2 (d) 21
  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) 28 (b) 29 (c) 21 (d) 22
  24. 24.A bag has 6 green 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) 7 (b) 18 (c) 13 (d) 2
  25. 25.A drawer has lots of 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) 3
  26. 26.A drawer has lots of 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) 5 (b) 2 (c) 3 (d) 4
  27. 27.A bag has 7 white balls and 4 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 2 (b) 9 (c) 8 (d) 11
  28. 28.A drawer has lots of black, white, 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) 4 (c) 2 (d) 5
  29. 29.A bag has 12 red balls and 9 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 21 (b) 13 (c) 14 (d) 2
  30. 30.A drawer has lots of black 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) 4 (b) 3 (c) 2 (d) 5
  31. 31.A bag has 5 black balls and 11 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 2 (b) 16 (c) 7 (d) 6
  32. 32.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) 37 (b) 24 (c) 36 (d) 25
  33. 33.A drawer has lots of 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) 5 (b) 2 (c) 8 (d) 4
  34. 34.A bag has 11 blue 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) 18 (b) 8 (c) 12 (d) 2
  35. 35.A bag has 6 red balls and 8 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 2 (b) 7 (c) 14 (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 day of the week? (a) 14 (b) 15 (c) 8 (d) 7
  37. 37.A bag has 3 white 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) 4 (b) 2 (c) 11 (d) 9
  38. 38.A drawer has lots of green, 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) 3 (d) 6
  39. 39.A bag has 12 blue balls and 3 yellow 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) 15 (c) 13 (d) 2
  40. 40.A bag has 8 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) 2 (b) 4 (c) 9 (d) 11

Answer key

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

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