May 8, 2007 8

## Math Brain Teaser: Concentric Shapes or The Unkindest Cut of All, Part 2 of 2

By: Caroline Latham

If you missed Part 1, also written by puzzle master Wes Carroll, you can start there and then come back here to Part 2.

**Concentric Shapes:
The Unkindest Cut of All, Part 2 of 2**

**Difficulty:** HARDER

**Type:** MATH (Spatial)

**Question:**

Imagine a square within a circle within a square. The circle just grazes each square at exactly four points. Find the ratio of the area of the larger square to the smaller.

In this puzzle you are working out many of the same skills as in Part I: spatial visualization (occipital lobes), memory (temporal lobes), logic (frontal lobes), planning (frontal lobes), and hypothesis generation (frontal lobes).

**Solution**:

Two to one.

**Explanation**:

Draw the smaller square’s diagonal to see that the the smaller square’s diagonal is the diameter of the circle. Divide the larger square into two equal rectangular halves to see that the larger square’s side is also the diameter of the circle. This means that the smaller square’s diagonal equals the larger square’s side. (Or, if you prefer, simply rotate the inner square by 45 degrees.) As we’ve seen in the earlier puzzle “The Unkindest Cut Of All,” the area of the smaller square is half that of the larger, making the ratio two to one.

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