Cats & Dogs #225: 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 R7C7 and R7C6 means they are different. R7C7 is a dog, so R7C6 is a cat.
  2. The × clue between R7C7 and R7C8 means they are different. R7C7 is a dog, so R7C8 is a cat.
  3. R8C5 and R10C5 are both cats with one square between them, so R9C5 must be a dog.
  4. Imagine R3C6 were a dog: following the rules from there soon forces a contradiction, so R3C6 must be a cat.
  5. R3C6 and R3C8 are both cats with one square between them, so R3C7 must be a dog.
  6. Imagine R4C5 were a cat: following the rules from there soon forces a contradiction, so R4C5 must be a dog.
  7. The = clue between R4C5 and R4C6 means they are the same. R4C5 is a dog, so R4C6 is a dog.
  8. R4C5 and R4C6 are both dogs, so R4C4 must be a cat (no three in a row).
  9. The = clue between R4C4 and R3C4 means they are the same. R4C4 is a cat, so R3C4 is a cat.
  10. R3C4 and R3C6 are both cats with one square between them, so R3C5 must be a dog.
  11. R4C5 and R4C6 are both dogs, so R4C7 must be a cat (no three in a row).
  12. R3C4 and R4C4 are both cats, so R2C4 must be a dog (no three in a row).
  13. The × clue between R2C4 and R1C4 means they are different. R2C4 is a dog, so R1C4 is a cat.
  14. R1C3 and R1C4 are both cats, so R1C2 must be a dog (no three in a row).
  15. R1C3 and R1C4 are both cats, so R1C5 must be a dog (no three in a row).
  16. R1C5 and R1C6 are both dogs, so R1C7 must be a cat (no three in a row).
  17. R3C4 and R4C4 are both cats, so R5C4 must be a dog (no three in a row).
  18. The = clue between R5C4 and R5C3 means they are the same. R5C4 is a dog, so R5C3 is a dog.
  19. The = clue between R5C3 and R4C3 means they are the same. R5C3 is a dog, so R4C3 is a dog.
  20. R5C3 and R5C4 are both dogs, so R5C2 must be a cat (no three in a row).
  21. R5C3 and R5C4 are both dogs, so R5C5 must be a cat (no three in a row).
  22. R4C3 and R5C3 are both dogs, so R3C3 must be a cat (no three in a row).
  23. R3C3 and R3C4 are both cats, so R3C2 must be a dog (no three in a row).
  24. R1C2 and R3C2 are both dogs with one square between them, so R2C2 must be a cat.
  25. R1C3 and R3C3 are both cats with one square between them, so R2C3 must be a dog.
  26. R2C3 and R2C4 are both dogs, so R2C5 must be a cat (no three in a row).
  27. R4C3 and R5C3 are both dogs, so R6C3 must be a cat (no three in a row).
  28. The = clue between R6C3 and R7C3 means they are the same. R6C3 is a cat, so R7C3 is a cat.
  29. R6C3 and R7C3 are both cats, so R8C3 must be a dog (no three in a row).
  30. R4C6 and R5C6 are both dogs, so R6C6 must be a cat (no three in a row).
  31. R6C6 and R7C6 are both cats, so R8C6 must be a dog (no three in a row).
  32. Imagine R2C6 were a dog: following the rules from there soon forces a contradiction, so R2C6 must be a cat.
  33. R2C5 and R2C6 are both cats, so R2C7 must be a dog (no three in a row).
  34. Imagine R7C2 were a cat: following the rules from there soon forces a contradiction, so R7C2 must be a dog.
  35. Imagine R9C2 were a dog: following the rules from there soon forces a contradiction, so R9C2 must be a cat.
  36. The = clue between R9C2 and R10C2 means they are the same. R9C2 is a cat, so R10C2 is a cat.
  37. R9C2 and R10C2 are both cats, so R8C2 must be a dog (no three in a row).
  38. R8C2 and R8C3 are both dogs, so R8C1 must be a cat (no three in a row).
  39. R8C2 and R8C3 are both dogs, so R8C4 must be a cat (no three in a row).
  40. R7C2 and R8C2 are both dogs, so R6C2 must be a cat (no three in a row).
  41. R6C2 and R6C3 are both cats, so R6C1 must be a dog (no three in a row).
  42. R6C2 and R6C3 are both cats, so R6C4 must be a dog (no three in a row).
  43. R5C2 and R6C2 are both cats, so R4C2 must be a dog (no three in a row).
  44. R4C2 and R4C3 are both dogs, so R4C1 must be a cat (no three in a row).
  45. R5C4 and R6C4 are both dogs, so R7C4 must be a cat (no three in a row).
  46. R7C3 and R7C4 are both cats, so R7C5 must be a dog (no three in a row).
  47. R7C4 and R8C4 are both cats, so R9C4 must be a dog (no three in a row).
  48. R9C4 and R9C5 are both dogs, so R9C3 must be a cat (no three in a row).
  49. R9C2 and R9C3 are both cats, so R9C1 must be a dog (no three in a row).
  50. R9C4 and R9C5 are both dogs, so R9C6 must be a cat (no three in a row).
  51. The × clue between R9C6 and R10C6 means they are different. R9C6 is a cat, so R10C6 is a dog.
  52. Column 3 already has its 5 cats, so R10C3 must be dogs.
  53. Column 4 already has its 5 cats, so R10C4 must be dogs.
  54. Column 5 already has its 5 dogs, so R6C5 must be cats.
  55. R6C5 and R6C6 are both cats, so R6C7 must be a dog (no three in a row).
  56. R6C7 and R7C7 are both dogs, so R5C7 must be a cat (no three in a row).
  57. R6C7 and R7C7 are both dogs, so R8C7 must be a cat (no three in a row).
  58. Imagine R3C1 were a cat: following the rules from there soon forces a contradiction, so R3C1 must be a dog.
  59. Imagine R4C10 were a dog: following the rules from there soon forces a contradiction, so R4C10 must be a cat.
  60. Imagine R5C1 were a dog: following the rules from there soon forces a contradiction, so R5C1 must be a cat.
  61. Imagine R2C8 were a cat: following the rules from there soon forces a contradiction, so R2C8 must be a dog.
  62. R2C7 and R2C8 are both dogs, so R2C9 must be a cat (no three in a row).
  63. Imagine R4C8 were a dog: following the rules from there soon forces a contradiction, so R4C8 must be a cat.
  64. R4C7 and R4C8 are both cats, so R4C9 must be a dog (no three in a row).
  65. The × clue between R4C9 and R5C9 means they are different. R4C9 is a dog, so R5C9 is a cat.
  66. The × clue between R5C9 and R5C10 means they are different. R5C9 is a cat, so R5C10 is a dog.
  67. R5C7 and R5C9 are both cats with one square between them, so R5C8 must be a dog.
  68. Imagine R6C8 were a cat: following the rules from there soon forces a contradiction, so R6C8 must be a dog.
  69. R6C7 and R6C8 are both dogs, so R6C9 must be a cat (no three in a row).
  70. R5C9 and R6C9 are both cats, so R7C9 must be a dog (no three in a row).
  71. The × clue between R7C9 and R7C10 means they are different. R7C9 is a dog, so R7C10 is a cat.
  72. The = clue between R7C10 and R8C10 means they are the same. R7C10 is a cat, so R8C10 is a cat.
  73. R7C10 and R8C10 are both cats, so R6C10 must be a dog (no three in a row).
  74. R7C10 and R8C10 are both cats, so R9C10 must be a dog (no three in a row).
  75. Row 7 already has its 5 cats, so R7C1 must be dogs.
  76. Row 8 already has its 5 cats, so R8C8 and R8C9 must be dogs.
  77. R7C9 and R8C9 are both dogs, so R9C9 must be a cat (no three in a row).
  78. Imagine R1C8 were a cat: following the rules from there soon forces a contradiction, so R1C8 must be a dog.
  79. Column 8 already has its 5 dogs, so R9C8 must be cats.
  80. R9C6 and R9C8 are both cats with one square between them, so R9C7 must be a dog.
  81. Column 7 already has its 5 dogs, so R10C7 must be cats.
  82. Column 8 already has its 5 dogs, so R10C8 must be cats.
  83. R10C7 and R10C8 are both cats, so R10C9 must be a dog (no three in a row).
  84. Row 10 already has its 5 dogs, so R10C1 must be cats.
  85. Imagine R1C1 were a cat: following the rules from there soon forces a contradiction, so R1C1 must be a dog.
  86. R1C1 and R3C1 are both dogs with one square between them, so R2C1 must be a cat.
  87. Row 1 already has its 5 dogs, so R1C9 must be cats.
  88. The = clue between R1C9 and R1C10 means they are the same. R1C9 is a cat, so R1C10 is a cat.
  89. R1C9 and R2C9 are both cats, so R3C9 must be a dog (no three in a row).
  90. Row 2 already has its 5 cats, so R2C10 must be dogs.
  91. Row 3 already has its 5 dogs, so R3C10 must be cats.

About this puzzle

This 10×10 grid starts with 8 filled squares and 15 = / × 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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