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Pigeonhole Principle Worksheet for Maths Olympiad

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

Answer key

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

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