Cats & Dogs #273: 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

Use pure logic to place every cat and dog. Each row and column holds 5 of each animal, three in a row is never allowed, and the clues between squares settle the rest.

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

About this puzzle

This 10×10 grid starts with 12 filled squares and 12 = / × clues. It can be solved in 85 steps using a "what if" trial of one square on top of the basic rules.

Whenever two of the same animal sit side by side, the squares on both ends must be the other animal. Look for these pairs first.

This kind of puzzle is known as a binary or Takuzu puzzle, here with cats and dogs and extra = / × clues.

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