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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 656 · 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 two of them were born in the same month? (a) 12 (b) 25 (c) 13 (d) 24
  2. 2.A drawer has lots of red, white, 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) 4 (b) 2 (c) 5 (d) 8
  3. 3.A drawer has lots of black, green, yellow, 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) 2 (b) 5 (c) 6 (d) 10
  4. 4.A drawer has lots of 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) 4 (b) 5 (c) 2 (d) 3
  5. 5.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) 14 (b) 15 (c) 21 (d) 22
  6. 6.A drawer has lots of blue, 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) 2 (b) 10 (c) 6 (d) 5
  7. 7.A bag has 6 green balls and 11 yellow 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) 17 (c) 2 (d) 7
  8. 8.A bag has 11 white balls and 12 green 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) 12 (c) 23 (d) 2
  9. 9.A drawer has lots of yellow, red 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) 4 (c) 2 (d) 3
  10. 10.A bag has 3 green balls and 4 red 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) 2 (d) 4
  11. 11.A drawer has lots of green, 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) 4 (b) 5 (c) 2 (d) 8
  12. 12.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) 36 (b) 25 (c) 37 (d) 24
  13. 13.A bag has 10 black 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) 18 (c) 11 (d) 2
  14. 14.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) 2 (c) 4 (d) 8
  15. 15.A bag has 10 black 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) 2 (b) 13 (c) 11 (d) 4
  16. 16.A bag has 7 green 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) 15 (b) 8 (c) 9 (d) 2
  17. 17.A bag has 8 green 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) 2 (b) 9 (c) 19 (d) 12
  18. 18.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) 4 (b) 2 (c) 3 (d) 6
  19. 19.A drawer has lots of white, 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) 4 (b) 8 (c) 2 (d) 5
  20. 20.A bag has 4 green balls and 6 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 2 (b) 5 (c) 6 (d) 10
  21. 21.A drawer has lots of 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) 2 (c) 6 (d) 3
  22. 22.A drawer has lots of black, 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) 6 (b) 2 (c) 10 (d) 5
  23. 23.A drawer has lots of 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) 3 (b) 4 (c) 2 (d) 6
  24. 24.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) 49 (c) 48 (d) 36
  25. 25.A bag has 10 green balls and 6 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 16 (b) 11 (c) 2 (d) 12
  26. 26.A drawer has lots of black, green, white, 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) 6 (c) 10 (d) 5
  27. 27.A drawer has lots of yellow, blue, 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) 2 (b) 8 (c) 5 (d) 4
  28. 28.A drawer has lots of blue, green, 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) 4 (c) 8 (d) 2
  29. 29.A bag has 3 yellow 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) 13 (c) 4 (d) 2
  30. 30.A bag has 5 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) 11 (b) 2 (c) 6 (d) 7
  31. 31.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) 7 (d) 8
  32. 32.A drawer has lots of blue, 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) 10 (b) 6 (c) 5 (d) 2
  33. 33.A bag has 4 white balls and 5 green balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 6 (b) 2 (c) 5 (d) 9
  34. 34.A drawer has lots of white, red, 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) 4 (c) 5 (d) 2
  35. 35.A drawer has lots of 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) 2 (b) 6 (c) 4 (d) 3
  36. 36.A drawer has lots of yellow, white, 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) 2 (b) 6 (c) 5 (d) 10
  37. 37.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) 3 (c) 4 (d) 2
  38. 38.A bag has 12 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) 2 (b) 10 (c) 21 (d) 13
  39. 39.A bag has 7 black balls and 5 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 8 (b) 9 (c) 2 (d) 12
  40. 40.A bag has 8 red balls and 6 black balls. Without looking, how many balls must you take out to be sure of getting 2 black balls? (a) 14 (b) 2 (c) 10 (d) 9

Answer key

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

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