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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 43 · 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 3 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) 15 (b) 2 (c) 4 (d) 13
  2. 2.A bag has 5 white balls and 9 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 7 (b) 6 (c) 14 (d) 2
  3. 3.A drawer has lots of blue, white, green, black 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) 10 (c) 2 (d) 5
  4. 4.A bag has 7 black balls and 10 green 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) 11 (d) 2
  5. 5.A bag has 6 red balls and 4 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) 10
  6. 6.A drawer has lots of red, 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) 4 (c) 8 (d) 5
  7. 7.A bag has 7 red balls and 8 black 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) 9 (c) 8 (d) 2
  8. 8.A bag has 7 blue balls and 6 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) 13 (d) 8
  9. 9.A drawer has lots of red, green, 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) 6 (b) 10 (c) 2 (d) 5
  10. 10.A bag has 3 blue balls and 3 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 6 (b) 4 (c) 5 (d) 2
  11. 11.A bag has 7 white balls and 10 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 9 (b) 8 (c) 17 (d) 2
  12. 12.A drawer has lots of red, 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) 2 (b) 3 (c) 4 (d) 6
  13. 13.A bag has 11 red balls and 12 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) 12 (c) 23 (d) 13
  14. 14.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) 15 (b) 14 (c) 8 (d) 7
  15. 15.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) 36 (b) 48 (c) 37 (d) 49
  16. 16.A bag has 10 white balls and 3 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 13 (b) 12 (c) 11 (d) 2
  17. 17.A drawer has lots of 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) 2 (b) 3 (c) 6 (d) 4
  18. 18.A bag has 12 red balls and 6 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 14 (b) 2 (c) 18 (d) 13
  19. 19.A drawer has lots of green, red, 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) 8 (b) 2 (c) 4 (d) 5
  20. 20.A drawer has lots of white, red, yellow, 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) 2 (b) 10 (c) 6 (d) 5
  21. 21.A bag has 7 white balls and 12 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 19 (b) 8 (c) 9 (d) 2
  22. 22.A drawer has lots of black, yellow, 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) 5 (b) 10 (c) 6 (d) 2
  23. 23.A bag has 4 white 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) 5 (b) 11 (c) 2 (d) 8
  24. 24.A bag has 9 green balls and 12 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 11 (b) 2 (c) 10 (d) 21
  25. 25.A bag has 8 red balls and 12 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 20 (b) 2 (c) 10 (d) 9
  26. 26.A bag has 10 yellow balls and 9 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 12 (b) 2 (c) 11 (d) 19
  27. 27.A bag has 9 red balls and 3 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 2 (b) 10 (c) 11 (d) 12
  28. 28.A drawer has lots of yellow, 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) 3 (b) 6 (c) 2 (d) 4
  29. 29.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) 5 (b) 2 (c) 3 (d) 4
  30. 30.A drawer has lots of green, black, 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) 5 (b) 10 (c) 6 (d) 2
  31. 31.A bag has 3 green balls and 5 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 4 (b) 8 (c) 2 (d) 5
  32. 32.A drawer has lots of red, green, blue, 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) 5 (b) 2 (c) 6 (d) 10
  33. 33.A bag has 11 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) 14 (b) 4 (c) 12 (d) 2
  34. 34.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) 24 (b) 36 (c) 37 (d) 25
  35. 35.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) 3 (b) 2 (c) 4 (d) 6
  36. 36.A drawer has lots of white, blue, black, 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) 10 (b) 6 (c) 5 (d) 2
  37. 37.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) 4 (b) 3 (c) 5 (d) 2
  38. 38.A bag has 3 green 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) 13 (c) 11 (d) 4
  39. 39.A bag has 9 blue balls and 5 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 2 (b) 14 (c) 11 (d) 10
  40. 40.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) 2 (b) 3 (c) 5 (d) 4

Answer key

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

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