1001math: Geometric progressions
The puzzles present geometric progressions where the common ratio is a fraction, shown as an area on a diagram, with the sum of the series being the fraction of the shape that is shaded. While it is possible to arrive at the sums using the formula for summing a geometric progression, each is accessible using logic.
The challenge is to find the fraction of the shape that is shaded. In this square each quarter has the same area, so we only need to focus on one of them. This is the upper right quarter.
This can be thought of as a series of L-shaped pieces that get smaller and smaller. While the Ls are all different in size, they each have the same fraction shaded:
Since one-third of each L is shaded, and the whole figure is composed of these Ls, then one-third of the big square is shaded.
Subtle alterations can make a big difference to the difficulty of the problem. Consider the diagram below:
As previously, it is helpful to consider one quarter of the shape.
The largest L is three-quarters of the shape and is two-thirds shaded. The remainder of the shape is one-quarter and as in the previous example one-third of it is shaded.
The calculation of the area becomes
We have recently added the complete collection of resources from Sian Zelbo’s ‘1001math’ website, the materials include investigations, puzzles, and other enrichment activities.
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