Permutations & Combinations
Permutation and Combination
star_batch_jee_advanced_2025
Grade 11
Question:
Consider a $6 \times 6$ square which is dissected into 9 rectangles by lines parallel to its sides such that all the rectangles have integral sides. What is the minimum number of congruent rectangles?
Step-by-Step Solution
Key Concept: The geometric constraint that parallel lines create a grid structure forces certain rectangles to repeat; achieving exactly 2 congruent rectangles is minimal while still satisfying the dissection requirements.
We need to dissect a $6 \times 6$ square into 9 rectangles with integral sides, minimizing congruent pairs. If all rectangles were distinct, we'd need the sum of their areas to equal 36. The key constraint is that rectangles must tile perfectly using lines parallel to sides. By systematically constructing tilings, we can achieve 7 distinct rectangles with 2 pairs of congruent rectangles (meaning 2 rectangles appear twice each, totaling 9 rectangles). This gives a minimum of 2 congruent rectangles. Attempting to have fewer than 2 congruent rectangles (all 9 distinct) leads to contradictions in the tiling constraints imposed by parallel lines creating integral dimensions.
Correct Answer: 2