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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 169 · 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.

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

Answer key

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

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