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Jan 13, 2024 •240 likes •2.23k views

Fun With Number Systems Or: How Knowing Number Systems Can Help You Interview Better Finding the Odd Ball 1 2 9 10 11 3 3 2 1 10 11 9 Goal: Find the odd ball and whether it's heavier or lighter in three weighings. The Solution 5 6

  1. Example: Three Weighings Heavier Lighter -1 00- 1 00+ -2 0-+ 2 0+- -3 0-0 3 0+0 -4 0-- 4 0++ -5 -++ 5 +-- -6 -+0 6 +-0 -7 -+- 7 +-+ 8 +0- -8 -0+ 9 +00 -9 -00 10 +0+ -10 -0- -11 --+ 11 ++- 12 ++0 -12 --0

  2. Example: Three Weighings Heavier Lighter -1 00- 1 00+ -2 0-+ 2 0+- -3 0-0 3 0+0 -4 0-- 4 0++ -5 -++ 5 +-- -6 -+0 6 +-0 -7 -+- 7 +-+ 8 +0- -8 -0+ 9 +00 -9 -00 10 +0+ -10 -0- -11 --+ 11 ++- 12 ++0 -12 --0

  3. Example: Four Weighings 1 000+ 14 +--- 27 +000 2 00+- 15 +--0 28 +00+ 3 00+0 16 +--+ 29 +0+- 4 00++ 17 +-0- 30 +0+0 5 0+-- 18 +-00 31 +0++ 6 0+-0 19 +-0+ 32 ++-- 7 0+-+ 20 +-+- 33 ++-0 8 0+0- 21 +-+0 34 ++-+ 9 0+00 22 +-++ 35 ++0- 10 0+0+ 23 +0-- 36 ++00 11 0++- 24 +0-0 37 ++0+ 12 0++0 25 +0-+ 38 +++- 13 0+++ 26 +00- 39 +++0 40 ++++

  4. Example: Four Weighings 1 000+ 14 +--- 27 +000 2 00+- 15 +--0 28 +00+ 3 00+0 16 +--+ 29 +0+- 4 00++ 17 +-0- 30 +0+0 5 0+-- 18 +-00 31 +0++ 6 0+-0 19 +-0+ 32 ++-- 7 0+-+ 20 +-+- 33 ++-0 8 0+0- 21 +-+0 34 ++-+ 9 0+00 22 +-++ 35 ++0- 10 0+0+ 23 +0-- 36 ++00 11 0++- 24 +0-0 37 ++0+ 12 0++0 25 +0-+ 38 +++- 13 0+++ 26 +00- 39 +++0 40 ++++

  5. Example: Four Weighings 1 000+ 14 +--- 27 +000 2 00+- 15 +--0 28 +00+ 3 00+0 16 +--+ 29 +0+- 4 00++ 17 +-0- 30 +0+0 5 0+-- 18 +-00 31 +0++ 6 0+-0 19 +-0+ 32 ++-- 7 0+-+ 20 +-+- 33 ++-0 8 0+0- 21 +-+0 34 ++-+ 9 0+00 22 +-++ 35 ++0- 10 0+0+ 23 +0-- 36 ++00 11 0++- 24 +0-0 37 ++0+ 12 0++0 25 +0-+ 38 +++- 13 0+++ 26 +00- 39 +++0

  6. Example: Four Weighings 1 000+ 14 +--- 27 +000 2 00+- 15 +--0 28 +00+ 3 00+0 16 +--+ 29 +0+- 4 00++ 17 +-0- 30 +0+0 5 0+-- 18 +-00 31 +0++ 6 0+-0 19 +-0+ 32 ++-- 7 0+-+ 20 +-+- 33 ++-0 8 0+0- 21 +-+0 34 ++-+ 9 0+00 22 +-++ 35 ++0- 10 0+0+ 23 +0-- 36 ++00 11 0++- 24 +0-0 37 ++0+ 12 0++0 38 +++- 25 +0-+ 13 0+++ 26 +00- 39 +++0

  7. Example: Four Weighings 1 000+ 14 +--- -27 -000 2 00+- 15 +--0 -28 -00- 3 00+0 16 +--+ -29 -0-+ 4 00++ 17 +-0- -30 -0-0 5 0+-- 18 +-00 -31 -0-- 6 0+-0 19 +-0+ 32 ++-- 7 0+-+ 20 +-+- 33 ++-0 8 0+0- 21 +-+0 34 ++-+ 9 0+00 22 +-++ -35 --0+ 10 0+0+ -23 -0++ -36 --00 11 0++- -24 -0+0 -37 --0- 12 0++0 38 +++- -25 -0+- 13 0+++ -26 -00+ -39 ---0

  8. Example: Four Weighings 1 000+ 14 +--- -27 -000 2 00+- 15 +--0 -28 -00- 3 00+0 16 +--+ -29 -0-+ 4 00++ 17 +-0- -30 -0-0 5 0+-- 18 +-00 -31 -0-- 6 0+-0 19 +-0+ 32 ++-- 7 0+-+ 20 +-+- 33 ++-0 8 0+0- 21 +-+0 34 ++-+ 9 0+00 22 +-++ -35 --0+ 10 0+0+ -23 -0++ -36 --00 11 0++- -24 -0+0 -37 --0- 12 0++0 -25 -0+- 38 +++- 13 0+++ -26 -00+ -39 ---0

  9. Some Insights ● What number did we drop? ● With 2 trits, dropped ++. ● With 3 trits, dropped +++. ● With 4 trits, dropped ++++. ● Always drop ++…++ ● What numbers did we invert? ● With two trits: +0 ● With three trits: ++0, +0-, +00, +0- ● With four trits: +++0, ++0-, ++00, ++0+, +0--, +0-0, +0-+, +00-, +000, +00+, +0+-, +0+0, +0++ ● Always invert numbers starting with ++…++0. ● This always works!

  10. How Many Extra +s Per Column? ● Answer: The 3 j column has (3 j – 1) / 2 extra +'s. 0+ +- +0

  11. How Many Extra +s Per Column? ● Answer: The 3 j column has (3 j – 1) / 2 extra +'s. 00+ 0+- 0+0

  12. How Many Extra +s Per Column? ● Answer: The 3 j column has (3 j – 1) / 2 extra +'s. 00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++- ++0

  13. How Many Extra +s Per Column? ● Answer: The 3 j column has (3 j – 1) / 2 extra +'s. 00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++- ++0

  14. How Many +'s Get Flipped? ● Answer: The 3 j column has (3 j -1) / 2 flipped +'s.

  15. How Many +'s Get Flipped? ● Answer: The 3 j column has (3 j -1) / 2 flipped +'s. 0+ +- +0

  16. How Many +'s Get Flipped? ● Answer: The 3 j column has (3 j -1) / 2 flipped +'s. 0+ +- +0

  17. How Many +'s Get Flipped? ● Answer: The 3 j column has (3 j -1) / 2 flipped +'s. 00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++- ++0

  18. How Many +'s Get Flipped? ● Answer: The 3 j column has (3 j -1) / 2 flipped +'s. 00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++- ++0

  19. How Many +'s Get Flipped? ● Answer: The 3 j column has (3 j -1) / 2 flipped +'s. 00+ 0+- 0+0 0++ j − 1 j − 1 i = 3 ∑ +-- 3 +-0 2 i = 0 +-+ +0- +00 +0+ ++- ++0

  20. Summary ● A three-way scale lends itself naturally to a balanced ternary encoding for each of the balls. ● Given an encoding with the same number of +'s and -'s in each column, we can use the scale to read off one trit of the answer at a time. ● Flipping numbers starting with +0, ++0, etc. guarantees an encoding with this property.

  21. Generating Permutations

  22. “You are given a sorted string S of unique characters. Write a Java-style iterator that traverses all the permutations of S in lexicographical order.”

  23. Example

  24. Example abc

  25. Example abc acb bac bca cab cba

  26. Example 0 abc 1 acb 2 bac 3 bca 4 cab 5 cba

  27. Lehmer Codes B A E D C

  28. Lehmer Codes B A E D C

  29. Lehmer Codes B A E D C

  30. Lehmer Codes B A E D C 1

  31. Lehmer Codes B A E D C 1

  32. Lehmer Codes B A E D C 1

  33. Lehmer Codes B A E D C 1 0

  34. Lehmer Codes B A E D C 1 0

  35. Lehmer Codes B A E D C 1 0

  36. Lehmer Codes B A E D C 1 0

  37. Lehmer Codes B A E D C 1 0 2

  38. Lehmer Codes B A E D C 1 0 2

  39. Lehmer Codes B A E D C 1 0 2

  40. Lehmer Codes B A E D C 1 0 2

  41. Lehmer Codes B A E D C 1 0 2 1

  42. Lehmer Codes B A E D C 1 0 2 1

  43. Lehmer Codes B A E D C 1 0 2 1

  44. Lehmer Codes B A E D C 1 0 2 1 0

  45. Lehmer Codes B A E D C 1 0 2 1 0

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