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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 860 · 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) 21 (b) 22 (c) 29 (d) 28
  2. 2.A drawer has lots of red, 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) 8 (b) 2 (c) 4 (d) 5
  3. 3.A drawer has lots of 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) 2 (c) 3 (d) 4
  4. 4.A drawer has lots of green, yellow 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) 4 (c) 2 (d) 3
  5. 5.A bag has 4 red balls and 10 white 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) 2 (c) 11 (d) 14
  6. 6.A drawer has lots of green, red, blue, yellow 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) 5 (c) 2 (d) 10
  7. 7.What is the smallest number of children in a group that makes sure at least three of them were born in the same day of the week? (a) 15 (b) 22 (c) 14 (d) 21
  8. 8.A bag has 4 white balls and 12 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 5 (b) 6 (c) 2 (d) 16
  9. 9.A bag has 6 black balls and 9 green balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 10 (b) 15 (c) 2 (d) 7
  10. 10.A bag has 10 black balls and 4 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 11 (b) 14 (c) 2 (d) 12
  11. 11.A drawer has lots of black, red, yellow 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) 5 (c) 8 (d) 2
  12. 12.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) 3 (b) 6 (c) 4 (d) 2
  13. 13.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) 6 (c) 4 (d) 3
  14. 14.A bag has 8 yellow balls and 12 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) 20
  15. 15.A bag has 4 blue balls and 11 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 15 (b) 2 (c) 6 (d) 5
  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) 4 (c) 2 (d) 5
  17. 17.A drawer has lots of yellow, blue, red, 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) 2 (b) 10 (c) 6 (d) 5
  18. 18.A drawer has lots of white, 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) 6 (c) 4 (d) 3
  19. 19.A bag has 3 blue balls and 5 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 4 (b) 8 (c) 5 (d) 2
  20. 20.A bag has 3 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) 2 (b) 4 (c) 8 (d) 10
  21. 21.A bag has 10 white balls and 8 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 11 (b) 18 (c) 2 (d) 12
  22. 22.A bag has 11 black balls and 3 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) 4 (c) 12 (d) 14
  23. 23.A drawer has lots of yellow, blue, 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) 2 (b) 6 (c) 10 (d) 5
  24. 24.A bag has 7 yellow balls and 4 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 2 (b) 8 (c) 9 (d) 11
  25. 25.A bag has 5 red balls and 4 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 6 (b) 7 (c) 2 (d) 9
  26. 26.A bag has 8 green balls and 8 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 9 (b) 10 (c) 16 (d) 2
  27. 27.A drawer has lots of black, 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) 8 (d) 5
  28. 28.A bag has 6 black balls and 8 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) 7 (c) 14 (d) 9
  29. 29.A bag has 6 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) 9 (b) 14 (c) 7 (d) 2
  30. 30.A bag has 9 blue 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) 12 (b) 2 (c) 20 (d) 10
  31. 31.A bag has 3 white balls and 11 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 5 (b) 14 (c) 4 (d) 2
  32. 32.A bag has 11 black 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) 23 (b) 2 (c) 13 (d) 12
  33. 33.A bag has 3 green balls and 3 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 6 (b) 4 (c) 2 (d) 5
  34. 34.A drawer has lots of red, blue, 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) 4 (b) 5 (c) 2 (d) 8
  35. 35.A drawer has lots of yellow, red, white, 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) 5 (b) 2 (c) 10 (d) 6
  36. 36.A drawer has lots of red, yellow, green, 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) 10 (b) 2 (c) 6 (d) 5
  37. 37.A bag has 3 blue balls and 5 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) 8 (c) 2 (d) 6
  38. 38.A bag has 8 white balls and 7 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 10 (b) 2 (c) 15 (d) 9
  39. 39.A bag has 5 yellow balls and 7 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 6 (b) 7 (c) 2 (d) 12
  40. 40.A bag has 8 red balls and 4 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 9 (b) 12 (c) 10 (d) 2

Answer key

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

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