Cats & Dogs #217: 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

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 R3C8 and R3C7 means they are the same. R3C8 is a cat, so R3C7 is a cat.
  2. The = clue between R9C4 and R8C4 means they are the same. R9C4 is a dog, so R8C4 is a dog.
  3. R3C7 and R3C8 are both cats, so R3C6 must be a dog (no three in a row).
  4. R3C4 and R3C6 are both dogs with one square between them, so R3C5 must be a cat.
  5. R3C7 and R3C8 are both cats, so R3C9 must be a dog (no three in a row).
  6. R8C4 and R8C5 are both dogs, so R8C3 must be a cat (no three in a row).
  7. The × clue between R8C3 and R8C2 means they are different. R8C3 is a cat, so R8C2 is a dog.
  8. R8C4 and R8C5 are both dogs, so R8C6 must be a cat (no three in a row).
  9. The × clue between R8C6 and R9C6 means they are different. R8C6 is a cat, so R9C6 is a dog.
  10. R9C1 and R9C3 are both cats with one square between them, so R9C2 must be a dog.
  11. R9C4 and R9C6 are both dogs with one square between them, so R9C5 must be a cat.
  12. R8C2 and R9C2 are both dogs, so R7C2 must be a cat (no three in a row).
  13. R8C2 and R9C2 are both dogs, so R10C2 must be a cat (no three in a row).
  14. R8C3 and R9C3 are both cats, so R7C3 must be a dog (no three in a row).
  15. R8C3 and R9C3 are both cats, so R10C3 must be a dog (no three in a row).
  16. R8C4 and R9C4 are both dogs, so R7C4 must be a cat (no three in a row).
  17. R8C4 and R9C4 are both dogs, so R10C4 must be a cat (no three in a row).
  18. R10C4 and R10C5 are both cats, so R10C6 must be a dog (no three in a row).
  19. R3C5 and R4C5 are both cats, so R2C5 must be a dog (no three in a row).
  20. R3C5 and R4C5 are both cats, so R5C5 must be a dog (no three in a row).
  21. Imagine R1C5 were a cat: following the rules from there soon forces a contradiction, so R1C5 must be a dog.
  22. Imagine R1C6 were a dog: following the rules from there soon forces a contradiction, so R1C6 must be a cat.
  23. Imagine R2C6 were a dog: following the rules from there soon forces a contradiction, so R2C6 must be a cat.
  24. Imagine R3C2 were a dog: following the rules from there soon forces a contradiction, so R3C2 must be a cat.
  25. The = clue between R3C2 and R4C2 means they are the same. R3C2 is a cat, so R4C2 is a cat.
  26. R3C2 and R4C2 are both cats, so R2C2 must be a dog (no three in a row).
  27. The × clue between R2C2 and R2C1 means they are different. R2C2 is a dog, so R2C1 is a cat.
  28. R3C2 and R4C2 are both cats, so R5C2 must be a dog (no three in a row).
  29. Imagine R4C3 were a cat: following the rules from there soon forces a contradiction, so R4C3 must be a dog.
  30. Imagine R4C4 were a dog: following the rules from there soon forces a contradiction, so R4C4 must be a cat.
  31. The = clue between R4C4 and R5C4 means they are the same. R4C4 is a cat, so R5C4 is a cat.
  32. R4C4 and R4C5 are both cats, so R4C6 must be a dog (no three in a row).
  33. R4C4 and R5C4 are both cats, so R6C4 must be a dog (no three in a row).
  34. R3C6 and R4C6 are both dogs, so R5C6 must be a cat (no three in a row).
  35. The = clue between R5C6 and R5C7 means they are the same. R5C6 is a cat, so R5C7 is a cat.
  36. R5C6 and R5C7 are both cats, so R5C8 must be a dog (no three in a row).
  37. R3C7 and R5C7 are both cats with one square between them, so R4C7 must be a dog.
  38. R4C6 and R4C7 are both dogs, so R4C8 must be a cat (no three in a row).
  39. R3C8 and R4C8 are both cats, so R2C8 must be a dog (no three in a row).
  40. Column 4 already has its 5 cats, so R1C4 must be dogs.
  41. R1C4 and R1C5 are both dogs, so R1C3 must be a cat (no three in a row).
  42. Column 6 already has its 5 dogs, so R6C6 must be cats.
  43. Imagine R1C1 were a cat: following the rules from there soon forces a contradiction, so R1C1 must be a dog.
  44. Imagine R3C10 were a cat: following the rules from there soon forces a contradiction, so R3C10 must be a dog.
  45. Imagine R8C7 were a dog: following the rules from there soon forces a contradiction, so R8C7 must be a cat.
  46. R8C6 and R8C7 are both cats, so R8C8 must be a dog (no three in a row).
  47. Row 8 already has its 5 dogs, so R8C9 and R8C10 must be cats.
  48. R8C10 and R10C10 are both cats with one square between them, so R9C10 must be a dog.
  49. Imagine R7C9 were a cat: following the rules from there soon forces a contradiction, so R7C9 must be a dog.
  50. Imagine R10C1 were a cat: following the rules from there soon forces a contradiction, so R10C1 must be a dog.
  51. Imagine R2C3 were a dog: following the rules from there soon forces a contradiction, so R2C3 must be a cat.
  52. R1C3 and R2C3 are both cats, so R3C3 must be a dog (no three in a row).
  53. R3C3 and R4C3 are both dogs, so R5C3 must be a cat (no three in a row).
  54. Row 3 already has its 5 dogs, so R3C1 must be cats.
  55. Column 3 already has its 5 cats, so R6C3 must be dogs.
  56. The × clue between R6C3 and R6C2 means they are different. R6C3 is a dog, so R6C2 is a cat.
  57. R6C3 and R6C4 are both dogs, so R6C5 must be a cat (no three in a row).
  58. R6C5 and R6C6 are both cats, so R6C7 must be a dog (no three in a row).
  59. Column 2 already has its 5 cats, so R1C2 must be dogs.
  60. Column 5 already has its 5 cats, so R7C5 must be dogs.
  61. R7C5 and R7C6 are both dogs, so R7C7 must be a cat (no three in a row).
  62. R7C7 and R8C7 are both cats, so R9C7 must be a dog (no three in a row).
  63. R9C6 and R9C7 are both dogs, so R9C8 must be a cat (no three in a row).
  64. Row 9 already has its 5 dogs, so R9C9 must be cats.
  65. R8C9 and R9C9 are both cats, so R10C9 must be a dog (no three in a row).
  66. Imagine R1C7 were a dog: following the rules from there soon forces a contradiction, so R1C7 must be a cat.
  67. R1C6 and R1C7 are both cats, so R1C8 must be a dog (no three in a row).
  68. R1C7 and R3C7 are both cats with one square between them, so R2C7 must be a dog.
  69. R2C7 and R2C8 are both dogs, so R2C9 must be a cat (no three in a row).
  70. Row 1 already has its 5 dogs, so R1C9 and R1C10 must be cats.
  71. Row 2 already has its 5 cats, so R2C10 must be dogs.
  72. R2C10 and R3C10 are both dogs, so R4C10 must be a cat (no three in a row).
  73. R4C8 and R4C10 are both cats with one square between them, so R4C9 must be a dog.
  74. R3C9 and R4C9 are both dogs, so R5C9 must be a cat (no three in a row).
  75. Row 5 already has its 5 cats, so R5C1 must be dogs.
  76. R4C1 and R5C1 are both dogs, so R6C1 must be a cat (no three in a row).
  77. Row 5 already has its 5 cats, so R5C10 must be dogs.
  78. Column 1 already has its 5 dogs, so R7C1 must be cats.
  79. Column 7 already has its 5 cats, so R10C7 must be dogs.
  80. R10C6 and R10C7 are both dogs, so R10C8 must be a cat (no three in a row).
  81. Column 9 already has its 5 cats, so R6C9 must be dogs.
  82. R6C7 and R6C9 are both dogs with one square between them, so R6C8 must be a cat.
  83. Row 6 already has its 5 cats, so R6C10 must be dogs.
  84. R5C10 and R6C10 are both dogs, so R7C10 must be a cat (no three in a row).
  85. Row 7 already has its 5 cats, so R7C8 must be dogs.

About this puzzle

This 10×10 grid starts with 13 filled squares and 9 = / × clues. It can be solved in 85 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.

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

More hard puzzles