54. An infinite plane has to be tiled (without gaps or overlaps) using identical tiles, each in the shape of a regular n-sided polygon. How many values are possible for n? (1) 3 (2) 5 (3) 6 (4) None of these?

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54. An infinite plane has to be tiled (without gaps or overlaps) using...

Analysis:

To tile an infinite plane using identical tiles in the shape of a regular n-sided polygon, we need to find the values of n that satisfy the tiling condition (without gaps or overlaps).

Approach:

We can analyze the possibilities for different values of n and determine whether they satisfy the tiling condition.

Case 1: n = 3 (Triangle)
If we consider a regular triangle as the tile, it is not possible to tile an infinite plane without gaps or overlaps. This is because the angles of a regular triangle sum up to 180 degrees, and when we try to arrange the tiles, the total sum of angles will always be a multiple of 180 degrees, resulting in gaps or overlaps.

Case 2: n = 4 (Square)
If we consider a regular square as the tile, it is possible to tile an infinite plane without gaps or overlaps. We can arrange the squares in a grid-like pattern, with each square adjacent to four other squares.

Case 3: n = 5 (Pentagon)
If we consider a regular pentagon as the tile, it is not possible to tile an infinite plane without gaps or overlaps. This is because the angles of a regular pentagon sum up to 540 degrees, which is not a multiple of 360 degrees (required for a complete rotation around a point).

Case 4: n = 6 (Hexagon)
If we consider a regular hexagon as the tile, it is possible to tile an infinite plane without gaps or overlaps. We can arrange the hexagons in a honeycomb-like pattern, with each hexagon adjacent to three other hexagons.

Conclusion:

From the analysis above, we can see that the only possible values of n that satisfy the tiling condition (without gaps or overlaps) are 4 (Square) and 6 (Hexagon). Therefore, the answer is (4) None of these, as none of the given options (3, 5, 6) are the correct values.

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54. An infinite plane has to be tiled (without gaps or overlaps) using...

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