Cats & Dogs #282: 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. R3C6 and R3C8 are both dogs with one square between them, so R3C7 must be a cat.
  2. The × clue between R3C7 and R4C7 means they are different. R3C7 is a cat, so R4C7 is a dog.
  3. The = clue between R4C7 and R4C6 means they are the same. R4C7 is a dog, so R4C6 is a dog.
  4. R4C6 and R4C7 are both dogs, so R4C5 must be a cat (no three in a row).
  5. The × clue between R4C5 and R3C5 means they are different. R4C5 is a cat, so R3C5 is a dog.
  6. R3C5 and R3C6 are both dogs, so R3C4 must be a cat (no three in a row).
  7. The × clue between R3C4 and R4C4 means they are different. R3C4 is a cat, so R4C4 is a dog.
  8. R4C6 and R4C7 are both dogs, so R4C8 must be a cat (no three in a row).
  9. R3C6 and R4C6 are both dogs, so R2C6 must be a cat (no three in a row).
  10. The × clue between R2C6 and R2C7 means they are different. R2C6 is a cat, so R2C7 is a dog.
  11. R3C6 and R4C6 are both dogs, so R5C6 must be a cat (no three in a row).
  12. Imagine R3C3 were a cat: following the rules from there soon forces a contradiction, so R3C3 must be a dog.
  13. Imagine R4C2 were a dog: following the rules from there soon forces a contradiction, so R4C2 must be a cat.
  14. Imagine R4C3 were a dog: following the rules from there soon forces a contradiction, so R4C3 must be a cat.
  15. R4C3 and R5C3 are both cats, so R6C3 must be a dog (no three in a row).
  16. Imagine R9C5 were a cat: following the rules from there soon forces a contradiction, so R9C5 must be a dog.
  17. The × clue between R9C5 and R9C6 means they are different. R9C5 is a dog, so R9C6 is a cat.
  18. The = clue between R9C5 and R10C5 means they are the same. R9C5 is a dog, so R10C5 is a dog.
  19. The = clue between R10C5 and R10C6 means they are the same. R10C5 is a dog, so R10C6 is a dog.
  20. R10C3 and R10C5 are both dogs with one square between them, so R10C4 must be a cat.
  21. R9C5 and R10C5 are both dogs, so R8C5 must be a cat (no three in a row).
  22. Imagine R7C4 were a cat: following the rules from there soon forces a contradiction, so R7C4 must be a dog.
  23. The = clue between R7C4 and R8C4 means they are the same. R7C4 is a dog, so R8C4 is a dog.
  24. R7C4 and R8C4 are both dogs, so R6C4 must be a cat (no three in a row).
  25. R7C4 and R8C4 are both dogs, so R9C4 must be a cat (no three in a row).
  26. Imagine R6C5 were a cat: following the rules from there soon forces a contradiction, so R6C5 must be a dog.
  27. Imagine R7C5 were a dog: following the rules from there soon forces a contradiction, so R7C5 must be a cat.
  28. Imagine R7C6 were a cat: following the rules from there soon forces a contradiction, so R7C6 must be a dog.
  29. Imagine R8C7 were a cat: following the rules from there soon forces a contradiction, so R8C7 must be a dog.
  30. Imagine R9C7 were a dog: following the rules from there soon forces a contradiction, so R9C7 must be a cat.
  31. R9C6 and R9C7 are both cats, so R9C8 must be a dog (no three in a row).
  32. Imagine R8C10 were a dog: following the rules from there soon forces a contradiction, so R8C10 must be a cat.
  33. The × clue between R8C10 and R9C10 means they are different. R8C10 is a cat, so R9C10 is a dog.
  34. Imagine R2C4 were a cat: following the rules from there soon forces a contradiction, so R2C4 must be a dog.
  35. Imagine R2C5 were a dog: following the rules from there soon forces a contradiction, so R2C5 must be a cat.
  36. Imagine R3C1 were a dog: following the rules from there soon forces a contradiction, so R3C1 must be a cat.
  37. Imagine R5C2 were a cat: following the rules from there soon forces a contradiction, so R5C2 must be a dog.
  38. Imagine R6C2 were a dog: following the rules from there soon forces a contradiction, so R6C2 must be a cat.
  39. Imagine R7C7 were a dog: following the rules from there soon forces a contradiction, so R7C7 must be a cat.
  40. Imagine R8C6 were a dog: following the rules from there soon forces a contradiction, so R8C6 must be a cat.
  41. Imagine R8C3 were a dog: following the rules from there soon forces a contradiction, so R8C3 must be a cat.
  42. Imagine R8C1 were a cat: following the rules from there soon forces a contradiction, so R8C1 must be a dog.
  43. Imagine R5C1 were a dog: following the rules from there soon forces a contradiction, so R5C1 must be a cat.
  44. Imagine R1C4 were a dog: following the rules from there soon forces a contradiction, so R1C4 must be a cat.
  45. Column 4 already has its 5 cats, so R5C4 must be dogs.
  46. Imagine R1C1 were a dog: following the rules from there soon forces a contradiction, so R1C1 must be a cat.
  47. R1C1 and R3C1 are both cats with one square between them, so R2C1 must be a dog.
  48. Imagine R1C3 were a cat: following the rules from there soon forces a contradiction, so R1C3 must be a dog.
  49. R1C3 and R3C3 are both dogs with one square between them, so R2C3 must be a cat.
  50. Imagine R1C5 were a dog: following the rules from there soon forces a contradiction, so R1C5 must be a cat.
  51. The × clue between R1C5 and R1C6 means they are different. R1C5 is a cat, so R1C6 is a dog.
  52. R1C6 and R1C7 are both dogs, so R1C8 must be a cat (no three in a row).
  53. The × clue between R1C8 and R1C9 means they are different. R1C8 is a cat, so R1C9 is a dog.
  54. Row 1 already has its 5 dogs, so R1C10 must be cats.
  55. Column 5 already has its 5 cats, so R5C5 must be dogs.
  56. Column 6 already has its 5 dogs, so R6C6 must be cats.
  57. Imagine R2C2 were a cat: following the rules from there soon forces a contradiction, so R2C2 must be a dog.
  58. R1C2 and R2C2 are both dogs, so R3C2 must be a cat (no three in a row).
  59. Imagine R2C8 were a dog: following the rules from there soon forces a contradiction, so R2C8 must be a cat.
  60. Imagine R2C9 were a dog: following the rules from there soon forces a contradiction, so R2C9 must be a cat.
  61. R2C8 and R2C9 are both cats, so R2C10 must be a dog (no three in a row).
  62. The = clue between R2C10 and R3C10 means they are the same. R2C10 is a dog, so R3C10 is a dog.
  63. R3C8 and R3C10 are both dogs with one square between them, so R3C9 must be a cat.
  64. R2C9 and R3C9 are both cats, so R4C9 must be a dog (no three in a row).
  65. R2C10 and R3C10 are both dogs, so R4C10 must be a cat (no three in a row).
  66. R4C10 and R5C10 are both cats, so R6C10 must be a dog (no three in a row).
  67. Column 10 already has its 5 cats, so R7C10 must be dogs.
  68. Imagine R5C7 were a dog: following the rules from there soon forces a contradiction, so R5C7 must be a cat.
  69. R5C6 and R5C7 are both cats, so R5C8 must be a dog (no three in a row).
  70. R5C7 and R7C7 are both cats with one square between them, so R6C7 must be a dog.
  71. Row 5 already has its 5 cats, so R5C9 must be dogs.
  72. R4C9 and R5C9 are both dogs, so R6C9 must be a cat (no three in a row).
  73. The = clue between R6C9 and R6C8 means they are the same. R6C9 is a cat, so R6C8 is a cat.
  74. Row 6 already has its 5 cats, so R6C1 must be dogs.
  75. The × clue between R6C1 and R7C1 means they are different. R6C1 is a dog, so R7C1 is a cat.
  76. The = clue between R7C1 and R7C2 means they are the same. R7C1 is a cat, so R7C2 is a cat.
  77. R7C1 and R7C2 are both cats, so R7C3 must be a dog (no three in a row).
  78. R6C2 and R7C2 are both cats, so R8C2 must be a dog (no three in a row).
  79. Column 3 already has its 5 dogs, so R9C3 must be cats.
  80. R9C3 and R9C4 are both cats, so R9C2 must be a dog (no three in a row).
  81. R8C2 and R9C2 are both dogs, so R10C2 must be a cat (no three in a row).
  82. Row 9 already has its 5 cats, so R9C1 must be dogs.
  83. R8C1 and R9C1 are both dogs, so R10C1 must be a cat (no three in a row).
  84. Row 10 already has its 5 cats, so R10C8 must be dogs.
  85. R9C8 and R10C8 are both dogs, so R8C8 must be a cat (no three in a row).
  86. R8C8 and R8C10 are both cats with one square between them, so R8C9 must be a dog.
  87. R6C8 and R8C8 are both cats with one square between them, so R7C8 must be a dog.
  88. R7C8 and R7C10 are both dogs with one square between them, so R7C9 must be a cat.
  89. Row 10 already has its 5 cats, so R10C9 must be dogs.

About this puzzle

This 10×10 grid starts with 11 filled squares and 16 = / × clues. It can be solved in 89 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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