Geometric Twins | A

Question 1

Expression Engineer!

Draw lines and split the region consisting of white squares into 6 smaller congruent regions.

Question diagram 1
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Solution
Understand the Question
  • The grid is a 5×55 \times 5 square containing a total of 2525 smaller unit squares.
  • With 11 central colored square removed, there are 25−1=2425 - 1 = 24 white squares remaining.
  • To divide the white region into 66 congruent regions, every region must have the exact same shape, size, and number of squares: 246=4\dfrac{24}{6} = 4 squares each.

Step 1 · Calculate Region Size

Total number of squares in the grid:

Total squares=5×5=25\begin{aligned} \text{Total squares} &= 5 \times 5 \\[0.6em] &= 25 \end{aligned}

Subtracting the 11 central colored square gives the number of white squares:

White squares=25−1=24\begin{aligned} \text{White squares} &= 25 - 1 \\[0.6em] &= 24 \end{aligned}

Dividing into 66 congruent regions:

Squares per region=Total white squaresNumber of regions=246=4\begin{aligned} \text{Squares per region} &= \dfrac{\text{Total white squares}}{\text{Number of regions}} \\[0.6em] &= \dfrac{24}{6} \\[0.6em] &= 4 \end{aligned}

Step 2 · Divide the Grid into Congruent Regions

Question diagram

Each of the 66 congruent regions consists of 44 unit squares arranged in an identical L-shape (tetromino).

Answer

Each congruent region contains 44 squares, dividing the 2424 white squares into 66 congruent L-shaped regions.

Common Mistakes
  • Forgetting the Center Square: Dividing the full 2525 squares by 66 instead of first subtracting the 11 colored center square to get 2424.
  • Non-congruent Shapes: Creating regions that have equal area (44 squares each) but different shapes (e.g., mixing straight 4-square lines with L-shapes). Congruent figures must have both identical shape and identical size.
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