Cats & Dogs #293: 10x10 Grid Puzzle (Hard)

hard

Tap a square: 🐱 cat → 🐶 dog → clear

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🐱 5 cats and 🐶 5 dogs in every row and column = same   × different

A 10x10 logic grid: 5 cats and 5 dogs in every row and column, no three of a kind in a row, and a few = and × clues to point the way.

  • Exactly 5 cats and 5 dogs in every row and every column.
  • Never three of the same animal in a row, across or down.
  • An = between two squares means the same animal; a × means different animals.

Step-by-step solution

Squares are written as R<row>C<column>, counting from the top-left.

  1. The × clue between R6C8 and R5C8 means they are different. R6C8 is a cat, so R5C8 is a dog.
  2. R5C8 and R5C9 are both dogs, so R5C7 must be a cat (no three in a row).
  3. R5C8 and R5C9 are both dogs, so R5C10 must be a cat (no three in a row).
  4. R3C9 and R5C9 are both dogs with one square between them, so R4C9 must be a cat.
  5. Imagine R2C4 were a dog: following the rules from there soon forces a contradiction, so R2C4 must be a cat.
  6. The = clue between R2C4 and R3C4 means they are the same. R2C4 is a cat, so R3C4 is a cat.
  7. R2C4 and R3C4 are both cats, so R4C4 must be a dog (no three in a row).
  8. R4C4 and R4C6 are both dogs with one square between them, so R4C5 must be a cat.
  9. The × clue between R4C5 and R5C5 means they are different. R4C5 is a cat, so R5C5 is a dog.
  10. Imagine R8C9 were a cat: following the rules from there soon forces a contradiction, so R8C9 must be a dog.
  11. The = clue between R8C9 and R9C9 means they are the same. R8C9 is a dog, so R9C9 is a dog.
  12. R8C9 and R9C9 are both dogs, so R7C9 must be a cat (no three in a row).
  13. R8C9 and R9C9 are both dogs, so R10C9 must be a cat (no three in a row).
  14. The = clue between R10C9 and R10C8 means they are the same. R10C9 is a cat, so R10C8 is a cat.
  15. R10C8 and R10C9 are both cats, so R10C7 must be a dog (no three in a row).
  16. R10C5 and R10C7 are both dogs with one square between them, so R10C6 must be a cat.
  17. R8C6 and R10C6 are both cats with one square between them, so R9C6 must be a dog.
  18. Imagine R6C7 were a cat: following the rules from there soon forces a contradiction, so R6C7 must be a dog.
  19. Imagine R1C7 were a dog: following the rules from there soon forces a contradiction, so R1C7 must be a cat.
  20. The = clue between R1C7 and R2C7 means they are the same. R1C7 is a cat, so R2C7 is a cat.
  21. R1C7 and R2C7 are both cats, so R3C7 must be a dog (no three in a row).
  22. The × clue between R3C7 and R4C7 means they are different. R3C7 is a dog, so R4C7 is a cat.
  23. R3C7 and R3C9 are both dogs with one square between them, so R3C8 must be a cat.
  24. R4C7 and R4C9 are both cats with one square between them, so R4C8 must be a dog.
  25. Imagine R1C5 were a dog: following the rules from there soon forces a contradiction, so R1C5 must be a cat.
  26. R1C5 and R1C7 are both cats with one square between them, so R1C6 must be a dog.
  27. Imagine R6C9 were a dog: following the rules from there soon forces a contradiction, so R6C9 must be a cat.
  28. R6C8 and R6C9 are both cats, so R6C10 must be a dog (no three in a row).
  29. Imagine R2C10 were a dog: following the rules from there soon forces a contradiction, so R2C10 must be a cat.
  30. Imagine R7C7 were a cat: following the rules from there soon forces a contradiction, so R7C7 must be a dog.
  31. R6C7 and R7C7 are both dogs, so R8C7 must be a cat (no three in a row).
  32. R8C6 and R8C7 are both cats, so R8C5 must be a dog (no three in a row).
  33. R8C6 and R8C7 are both cats, so R8C8 must be a dog (no three in a row).
  34. R8C8 and R8C9 are both dogs, so R8C10 must be a cat (no three in a row).
  35. R8C5 and R10C5 are both dogs with one square between them, so R9C5 must be a cat.
  36. Column 7 already has its 5 cats, so R9C7 must be dogs.
  37. R9C6 and R9C7 are both dogs, so R9C8 must be a cat (no three in a row).
  38. Imagine R1C8 were a cat: following the rules from there soon forces a contradiction, so R1C8 must be a dog.
  39. R1C8 and R1C10 are both dogs with one square between them, so R1C9 must be a cat.
  40. The × clue between R1C9 and R2C9 means they are different. R1C9 is a cat, so R2C9 is a dog.
  41. Imagine R2C6 were a dog: following the rules from there soon forces a contradiction, so R2C6 must be a cat.
  42. R2C6 and R2C7 are both cats, so R2C8 must be a dog (no three in a row).
  43. Column 8 already has its 5 dogs, so R7C8 must be cats.
  44. R7C8 and R7C9 are both cats, so R7C10 must be a dog (no three in a row).
  45. Imagine R3C1 were a dog: following the rules from there soon forces a contradiction, so R3C1 must be a cat.
  46. Imagine R3C3 were a cat: following the rules from there soon forces a contradiction, so R3C3 must be a dog.
  47. Imagine R3C5 were a dog: following the rules from there soon forces a contradiction, so R3C5 must be a cat.
  48. R3C4 and R3C5 are both cats, so R3C6 must be a dog (no three in a row).
  49. R3C6 and R4C6 are both dogs, so R5C6 must be a cat (no three in a row).
  50. Row 3 already has its 5 dogs, so R3C10 must be cats.
  51. R2C10 and R3C10 are both cats, so R4C10 must be a dog (no three in a row).
  52. Column 10 already has its 5 dogs, so R9C10 must be cats.
  53. Imagine R9C3 were a cat: following the rules from there soon forces a contradiction, so R9C3 must be a dog.
  54. The = clue between R9C3 and R9C4 means they are the same. R9C3 is a dog, so R9C4 is a dog.
  55. The × clue between R9C4 and R10C4 means they are different. R9C4 is a dog, so R10C4 is a cat.
  56. R9C3 and R9C4 are both dogs, so R9C2 must be a cat (no three in a row).
  57. Row 9 already has its 5 dogs, so R9C1 must be cats.
  58. R9C1 and R10C1 are both cats, so R8C1 must be a dog (no three in a row).
  59. Row 10 already has its 5 cats, so R10C2 and R10C3 must be dogs.
  60. R9C3 and R10C3 are both dogs, so R8C3 must be a cat (no three in a row).
  61. Imagine R8C2 were a cat: following the rules from there soon forces a contradiction, so R8C2 must be a dog.
  62. Row 8 already has its 5 dogs, so R8C4 must be cats.
  63. Imagine R4C3 were a dog: following the rules from there soon forces a contradiction, so R4C3 must be a cat.
  64. Imagine R5C2 were a dog: following the rules from there soon forces a contradiction, so R5C2 must be a cat.
  65. The = clue between R5C2 and R5C3 means they are the same. R5C2 is a cat, so R5C3 is a cat.
  66. R5C2 and R5C3 are both cats, so R5C1 must be a dog (no three in a row).
  67. R5C2 and R5C3 are both cats, so R5C4 must be a dog (no three in a row).
  68. R4C3 and R5C3 are both cats, so R6C3 must be a dog (no three in a row).
  69. R4C4 and R5C4 are both dogs, so R6C4 must be a cat (no three in a row).
  70. R6C4 and R8C4 are both cats with one square between them, so R7C4 must be a dog.
  71. The × clue between R7C4 and R7C5 means they are different. R7C4 is a dog, so R7C5 is a cat.
  72. The × clue between R7C5 and R6C5 means they are different. R7C5 is a cat, so R6C5 is a dog.
  73. R6C5 and R6C7 are both dogs with one square between them, so R6C6 must be a cat.
  74. R5C6 and R6C6 are both cats, so R7C6 must be a dog (no three in a row).
  75. Imagine R1C2 were a dog: following the rules from there soon forces a contradiction, so R1C2 must be a cat.
  76. Imagine R1C1 were a cat: following the rules from there soon forces a contradiction, so R1C1 must be a dog.
  77. Row 1 already has its 5 dogs, so R1C3 must be cats.
  78. Imagine R2C1 were a cat: following the rules from there soon forces a contradiction, so R2C1 must be a dog.
  79. Imagine R2C2 were a dog: following the rules from there soon forces a contradiction, so R2C2 must be a cat.
  80. R2C2 and R2C4 are both cats with one square between them, so R2C3 must be a dog.
  81. Column 3 already has its 5 dogs, so R7C3 must be cats.
  82. The × clue between R7C3 and R7C2 means they are different. R7C3 is a cat, so R7C2 is a dog.
  83. The × clue between R7C2 and R7C1 means they are different. R7C2 is a dog, so R7C1 is a cat.
  84. R7C2 and R8C2 are both dogs, so R6C2 must be a cat (no three in a row).
  85. R5C2 and R6C2 are both cats, so R4C2 must be a dog (no three in a row).
  86. Row 4 already has its 5 dogs, so R4C1 must be cats.
  87. Row 6 already has its 5 cats, so R6C1 must be dogs.

About this puzzle

This 10×10 grid starts with 12 filled squares and 15 = / × clues. It can be solved in 87 steps using a "what if" trial of one square on top of the basic rules.

Count as you go. Once a row or column has all of one animal, every remaining empty square in it must be the other.

Every puzzle on this site has exactly one solution that can be reached by deduction. If you are guessing, look again at the rows and columns that are nearly full.

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