Saturday, March 14, 2009

A Pi-Day Puzzle

Zazzle.com Pi Stamp
The mathematical symbol for pi

If you haven't looked at your calendar yet, today is Pi Day. This is the day in which the month and day of the calendar is 3/14, the first three digits to the mathematical formula for pi (3.14...).

Before I go any further, I want to ask you a puzzle that relates to pi.

Suppose you could put a steel band tightly around the equator of Earth. Assume, for the sake of this puzzle, that the earth is perfectly round, without hills or valleys, and that the steel band would make a exact circle around Earth where it is touching the surface evenly. Then, you take a cutting torch, open up a gap in this band, and weld in exactly 1 extra meter (for non-metric readers, approximately 1 yard) of metal.

Question -- how high would this extra meter of material allow the band to be raised, evenly, throughout the entire circumference of the earth. If you haven't heard this question before, the answer, provided below, will likely amaze you.

I have searched through various websites trying to find stamps that portray the symbol for pi, but have came up virtually empty-handed. I have yet to find a postage stamp, issued by a postal authority, with such an image.

I did find several representations of pi on stamps designed by Zazzle.com. For readers who have never heard of Zazzle, the company markets metered stamps with custom images that are valid for postage in several countries. Technically the stamps are metered stamps -- the barcoding on the stamp is what identifies the stamp as valid postage for the United States Postal Service -- the picture is just an add-on. Customers can provide their own images for the pictorial image on the stamp or purchase them with pre-made images.

Zazzle makes up various designs and sells them. They have several images with pi on them, including the two represented here.

Zazzle.com Pi Stamp
Pi to 80 decimal places

Are there any pi stamps issued by postal agencies and not companies like Zazzle? Maybe readers can provide another example, but a quick search through websites dedicated to stamps featuring mathematical-related terms failed to turn up an example.


Now, back to the puzzle. The steel band would be raised approximately 1/6th of a meter (approximately 6 inches for non-metric readers) above the surface of the earth! And even more amazingly, it doesn't matter what the diameter of the round surface is -- from something as small as a pea or as large as the sun -- the result is the same -- the band will be raised about 1/6th of a meter larger than the object in all cases.

For those who might doubt this, here is how the solution is determined. My apologies to mathematicians everywhere!

The circumference of an item is equal to the diameter of the object multiplied by pi. When you add one meter of material to the steel band, you are adding 1 meter to the circumference. Since pi is approximately 3, then the one meter of extra circumference increases the diameter of the circle by about 1/3 meter -- 1/3 meter in diameter * pi (approx. 3) yields approximately 1 meter in circumference. So by adding 1 meter to the steel band, we are, in effect, increasing the diameter of the band by 1/3 meter. This yields a radius increase of 1/6th meter (diameter = 2 times the radius). Thus approximately 1/6th of a meter is how much the steel band is raised from surface at any one spot.

For those who prefer their numbers more precise, divide the 1 meter by an approximation of pi (3.141592...) and you get 0.318309... of a meter for the increase in diameter, which yields 0.159159... of a meter increase in the radius.


Happy Pi Day!



1 comment:

Dan said...

This is an interesting puzzle but the comment of 1/6 of a meter for all cases is a little confusing. It is 1/6 of a meter in this case, but for a more universal answer the correct phrase is 1/6 (of the given units of measure). More of a terminology or typo than anyone but it is a little confusing.