# Solution to puzzle 37: Five marbles

Consider two adjacent marbles, of radii a < b. We will show that b/a is a constant, whose value is dependent only upon the slope of the funnel wall.

The marbles are in contact with each other, and therefore the vertical distance between their centers is b + a.

The marbles are also in contact with the funnel wall. Since the slope of the funnel wall (in cross section) is a constant, the two green triangles are similar. Hence the horizontal distance from the center of each marble to the funnel wall is bc and ac, respectively, where c = sec(x) is a constant dependent upon the slope of the funnel wall. (x is the angle the funnel wall makes with the vertical.)

Let the slope of the funnel wall be m.

Then m = (b + a) / [(b − a)c].

Rearranging, b/a = (mc + 1)/(mc − 1).

Hence the ratio of the radii of adjacent marbles is a constant, dependent only upon the slope of the funnel wall. Let this constant be k.

In this case, we have 18 = 8k^{4}.

So k^{2} = 3/2.

Therefore the radius of the middle marble is 8 · (3/2) = 12mm.

## Remarks

Note that 12 is the geometric mean of 8 and 18. For any odd number of marbles in such a configuration, the radius of the middle marble is the geometric mean of the radii of the smallest and largest marbles.

Source: Mark Ganson

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