Cats & Dogs #251: 10x10 Grid Puzzle (Hard)

hard

Tap a square: 🐱 cat → 🐶 dog → clear

0 / 100 · 0:00
🐱 5 cats and 🐶 5 dogs in every row and column = same   × different

Fill the 10x10 grid with cats and dogs so that every row and every column has exactly 5 of each, and never more than two of the same animal side by side.

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

About this puzzle

This 10×10 grid starts with 13 filled squares and 11 = / × clues. It can be solved in 85 steps using a "what if" trial of one square on top of the basic rules.

Watch for a gap: if the squares on either side of an empty square match, the empty one must be different.

Stuck? Open the step-by-step solution below -- each step names the squares involved, so you can use it as a hint and finish the puzzle yourself.

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