Cats & Dogs #212: 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 R8C3 and R8C4 means they are different. R8C3 is a cat, so R8C4 is a dog.
  2. Imagine R2C2 were a cat: following the rules from there soon forces a contradiction, so R2C2 must be a dog.
  3. Imagine R4C2 were a dog: following the rules from there soon forces a contradiction, so R4C2 must be a cat.
  4. The = clue between R4C2 and R4C3 means they are the same. R4C2 is a cat, so R4C3 is a cat.
  5. R4C2 and R4C3 are both cats, so R4C4 must be a dog (no three in a row).
  6. R3C2 and R4C2 are both cats, so R5C2 must be a dog (no three in a row).
  7. Imagine R10C1 were a cat: following the rules from there soon forces a contradiction, so R10C1 must be a dog.
  8. The = clue between R10C1 and R10C2 means they are the same. R10C1 is a dog, so R10C2 is a dog.
  9. R10C1 and R10C2 are both dogs, so R10C3 must be a cat (no three in a row).
  10. The × clue between R10C3 and R10C4 means they are different. R10C3 is a cat, so R10C4 is a dog.
  11. The = clue between R10C4 and R10C5 means they are the same. R10C4 is a dog, so R10C5 is a dog.
  12. The × clue between R10C5 and R9C5 means they are different. R10C5 is a dog, so R9C5 is a cat.
  13. R10C4 and R10C5 are both dogs, so R10C6 must be a cat (no three in a row).
  14. R10C6 and R10C7 are both cats, so R10C8 must be a dog (no three in a row).
  15. R8C3 and R10C3 are both cats with one square between them, so R9C3 must be a dog.
  16. R8C4 and R10C4 are both dogs with one square between them, so R9C4 must be a cat.
  17. R9C4 and R9C5 are both cats, so R9C6 must be a dog (no three in a row).
  18. Row 10 already has its 5 dogs, so R10C9 and R10C10 must be cats.
  19. R9C10 and R10C10 are both cats, so R8C10 must be a dog (no three in a row).
  20. The = clue between R8C10 and R8C9 means they are the same. R8C10 is a dog, so R8C9 is a dog.
  21. R8C9 and R8C10 are both dogs, so R8C8 must be a cat (no three in a row).
  22. Imagine R1C3 were a cat: following the rules from there soon forces a contradiction, so R1C3 must be a dog.
  23. The = clue between R1C3 and R2C3 means they are the same. R1C3 is a dog, so R2C3 is a dog.
  24. R2C2 and R2C3 are both dogs, so R2C1 must be a cat (no three in a row).
  25. R2C2 and R2C3 are both dogs, so R2C4 must be a cat (no three in a row).
  26. R1C3 and R2C3 are both dogs, so R3C3 must be a cat (no three in a row).
  27. R3C2 and R3C3 are both cats, so R3C1 must be a dog (no three in a row).
  28. R3C2 and R3C3 are both cats, so R3C4 must be a dog (no three in a row).
  29. R3C1 and R4C1 are both dogs, so R5C1 must be a cat (no three in a row).
  30. R5C1 and R6C1 are both cats, so R7C1 must be a dog (no three in a row).
  31. R3C3 and R4C3 are both cats, so R5C3 must be a dog (no three in a row).
  32. R5C2 and R5C3 are both dogs, so R5C4 must be a cat (no three in a row).
  33. Imagine R1C1 were a dog: following the rules from there soon forces a contradiction, so R1C1 must be a cat.
  34. The × clue between R1C1 and R1C2 means they are different. R1C1 is a cat, so R1C2 is a dog.
  35. R1C2 and R1C3 are both dogs, so R1C4 must be a cat (no three in a row).
  36. Imagine R6C2 were a dog: following the rules from there soon forces a contradiction, so R6C2 must be a cat.
  37. R6C1 and R6C2 are both cats, so R6C3 must be a dog (no three in a row).
  38. R5C3 and R6C3 are both dogs, so R7C3 must be a cat (no three in a row).
  39. Imagine R6C4 were a dog: following the rules from there soon forces a contradiction, so R6C4 must be a cat.
  40. R6C4 and R6C5 are both cats, so R6C6 must be a dog (no three in a row).
  41. R5C4 and R6C4 are both cats, so R7C4 must be a dog (no three in a row).
  42. Imagine R6C7 were a cat: following the rules from there soon forces a contradiction, so R6C7 must be a dog.
  43. R6C6 and R6C7 are both dogs, so R6C8 must be a cat (no three in a row).
  44. R6C8 and R8C8 are both cats with one square between them, so R7C8 must be a dog.
  45. Row 6 already has its 5 cats, so R6C9 must be dogs.
  46. R6C9 and R8C9 are both dogs with one square between them, so R7C9 must be a cat.
  47. Row 6 already has its 5 cats, so R6C10 must be dogs.
  48. R6C10 and R8C10 are both dogs with one square between them, so R7C10 must be a cat.
  49. Imagine R7C2 were a dog: following the rules from there soon forces a contradiction, so R7C2 must be a cat.
  50. R6C2 and R7C2 are both cats, so R8C2 must be a dog (no three in a row).
  51. R8C2 and R10C2 are both dogs with one square between them, so R9C2 must be a cat.
  52. Imagine R4C8 were a cat: following the rules from there soon forces a contradiction, so R4C8 must be a dog.
  53. Imagine R5C7 were a dog: following the rules from there soon forces a contradiction, so R5C7 must be a cat.
  54. The = clue between R5C7 and R5C8 means they are the same. R5C7 is a cat, so R5C8 is a cat.
  55. R5C7 and R5C8 are both cats, so R5C6 must be a dog (no three in a row).
  56. R5C7 and R5C8 are both cats, so R5C9 must be a dog (no three in a row).
  57. R5C6 and R6C6 are both dogs, so R4C6 must be a cat (no three in a row).
  58. R5C6 and R6C6 are both dogs, so R7C6 must be a cat (no three in a row).
  59. R5C9 and R6C9 are both dogs, so R4C9 must be a cat (no three in a row).
  60. Row 7 already has its 5 cats, so R7C5 and R7C7 must be dogs.
  61. R6C7 and R7C7 are both dogs, so R8C7 must be a cat (no three in a row).
  62. R8C7 and R8C8 are both cats, so R8C6 must be a dog (no three in a row).
  63. R8C4 and R8C6 are both dogs with one square between them, so R8C5 must be a cat.
  64. R8C7 and R10C7 are both cats with one square between them, so R9C7 must be a dog.
  65. R9C6 and R9C7 are both dogs, so R9C8 must be a cat (no three in a row).
  66. R9C8 and R9C10 are both cats with one square between them, so R9C9 must be a dog.
  67. Row 8 already has its 5 dogs, so R8C1 must be cats.
  68. Row 9 already has its 5 cats, so R9C1 must be dogs.
  69. Imagine R1C5 were a cat: following the rules from there soon forces a contradiction, so R1C5 must be a dog.
  70. Imagine R1C6 were a dog: following the rules from there soon forces a contradiction, so R1C6 must be a cat.
  71. The × clue between R1C6 and R1C7 means they are different. R1C6 is a cat, so R1C7 is a dog.
  72. Imagine R1C8 were a cat: following the rules from there soon forces a contradiction, so R1C8 must be a dog.
  73. R1C7 and R1C8 are both dogs, so R1C9 must be a cat (no three in a row).
  74. Row 1 already has its 5 dogs, so R1C10 must be cats.
  75. Imagine R2C5 were a cat: following the rules from there soon forces a contradiction, so R2C5 must be a dog.
  76. R1C5 and R2C5 are both dogs, so R3C5 must be a cat (no three in a row).
  77. Imagine R2C6 were a dog: following the rules from there soon forces a contradiction, so R2C6 must be a cat.
  78. The = clue between R2C6 and R2C7 means they are the same. R2C6 is a cat, so R2C7 is a cat.
  79. R2C6 and R2C7 are both cats, so R2C8 must be a dog (no three in a row).
  80. R1C6 and R2C6 are both cats, so R3C6 must be a dog (no three in a row).
  81. R1C8 and R2C8 are both dogs, so R3C8 must be a cat (no three in a row).
  82. Imagine R2C9 were a dog: following the rules from there soon forces a contradiction, so R2C9 must be a cat.
  83. R1C9 and R2C9 are both cats, so R3C9 must be a dog (no three in a row).
  84. Row 2 already has its 5 cats, so R2C10 must be dogs.
  85. Imagine R3C7 were a dog: following the rules from there soon forces a contradiction, so R3C7 must be a cat.
  86. R2C7 and R3C7 are both cats, so R4C7 must be a dog (no three in a row).
  87. Row 3 already has its 5 cats, so R3C10 must be dogs.
  88. R2C10 and R3C10 are both dogs, so R4C10 must be a cat (no three in a row).
  89. Row 4 already has its 5 cats, so R4C5 must be dogs.
  90. Column 5 already has its 5 dogs, so R5C5 must be cats.
  91. Row 5 already has its 5 cats, so R5C10 must be dogs.

About this puzzle

This 10×10 grid starts with 7 filled squares and 12 = / × clues. It can be solved in 91 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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