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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 343 · 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.

NameDateTime takenScore ___ / 40
  1. 1.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) 29 (b) 21 (c) 22 (d) 28
  2. 2.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) 48 (b) 36 (c) 49 (d) 37
  3. 3.A drawer has lots of blue, yellow, 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) 2 (b) 8 (c) 5 (d) 4
  4. 4.A drawer has lots of 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) 4 (c) 5 (d) 3
  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 month? (a) 25 (b) 13 (c) 24 (d) 12
  6. 6.A bag has 3 red balls and 7 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 10 (b) 5 (c) 4 (d) 2
  7. 7.A bag has 11 green balls and 12 red balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 23 (b) 2 (c) 13 (d) 12
  8. 8.A bag has 11 red 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) 14 (c) 4 (d) 12
  9. 9.A drawer has lots of black, white 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) 4 (c) 6 (d) 3
  10. 10.A drawer has lots of 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) 3 (c) 2 (d) 4
  11. 11.A drawer has lots of white, blue, 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) 5 (c) 8 (d) 2
  12. 12.A bag has 6 black balls and 6 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 2 (b) 12 (c) 7 (d) 8
  13. 13.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) 4 (b) 6 (c) 2 (d) 3
  14. 14.A bag has 4 white balls and 7 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 6 (b) 2 (c) 5 (d) 11
  15. 15.A bag has 3 red balls and 4 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 2 (b) 5 (c) 7 (d) 4
  16. 16.A drawer has lots of 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) 3 (b) 2 (c) 4 (d) 5
  17. 17.A drawer has lots of white, black, 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) 10 (b) 6 (c) 2 (d) 5
  18. 18.A bag has 10 black 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) 17 (b) 8 (c) 2 (d) 11
  19. 19.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
  20. 20.A bag has 3 blue balls and 11 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 4 (b) 5 (c) 2 (d) 14
  21. 21.A drawer has lots of 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) 5 (b) 2 (c) 3 (d) 4
  22. 22.A bag has 8 black balls and 11 white balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 19 (b) 2 (c) 12 (d) 9
  23. 23.A drawer has lots of black, 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) 5 (b) 4 (c) 8 (d) 2
  24. 24.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) 5 (b) 2 (c) 3 (d) 4
  25. 25.A drawer has lots of yellow, white, red, 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) 6 (b) 10 (c) 2 (d) 5
  26. 26.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) 2 (b) 4 (c) 5 (d) 3
  27. 27.A drawer has lots of 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) 3 (b) 6 (c) 4 (d) 2
  28. 28.A bag has 9 blue balls and 12 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 2 (b) 10 (c) 21 (d) 11
  29. 29.A bag has 11 white balls and 8 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 12 (b) 2 (c) 19 (d) 13
  30. 30.A bag has 6 yellow balls and 7 white 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) 8 (d) 13
  31. 31.A bag has 6 red 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) 17 (b) 7 (c) 2 (d) 12
  32. 32.A drawer has lots of blue, 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) 2 (b) 5 (c) 4 (d) 8
  33. 33.A drawer has lots of yellow, white, 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) 2 (b) 5 (c) 4 (d) 8
  34. 34.A bag has 3 blue 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) 4 (b) 10 (c) 2 (d) 12
  35. 35.A drawer has lots of 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) 3 (c) 2 (d) 4
  36. 36.A drawer has lots of red, 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) 8 (b) 2 (c) 5 (d) 4
  37. 37.A drawer has lots of red, yellow, green, 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) 6 (b) 10 (c) 5 (d) 2
  38. 38.A bag has 12 green 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) 13 (b) 19 (c) 8 (d) 2
  39. 39.A drawer has lots of blue, red, 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) 5 (b) 2 (c) 4 (d) 8
  40. 40.A bag has 10 blue 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) 2 (d) 11

Answer key

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

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