Cats & Dogs #245: 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

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

About this puzzle

This 10×10 grid starts with 6 filled squares and 16 = / × clues. It can be solved in 88 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.

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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