AUE: SDC 2002: Question 13 Proof of the Pudding

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Materials

The following materials were assembled for this experiment:

Four rolls each of 50 U.S.A. pennies, retrieved from their home under the sink where they had been accumulating in a stash over some number of years
Two right-angled metal brackets, obtained from a local purveyor of fine hardware
Measuring accoutrements and sundry other bits and pieces as deemed necessary to perform the experiments.

Methodology

The two metal "L" brackets were first tested for conformance to a 90 degree angle.

All experiments were conducted at room temperature, pressure, and humidity.

Then the brackets were nested to form three adjustable walls of a rectangle. Together with a suitably sized matchbox to serve as the fourth wall, this created a parallelepiped volume which was filled with stacks of pennies. These stacks were created to a sufficient height to comfortably exceed half the diameter of a penny.

The penny stacks were arranged in serried ranks, and an attempt was made to see if two more pennies could be slipped in edgewise at the end of one of the shorter ranks. See Fig 1.

Fig 1

It appears that there is enough room for two pennies (but not three) to fit into this space without apparent difficulty. This does not conform with the theoretical value of only 1.92 pennies fitting into this space. This may be on account of three reasons:

Theoretical calculations (which tell us that we have 8% excess penny thickness to fit in a second penny in the space—a veritable "red cent hair" as one wag put it) assume a penny with "square shoulders". It is at the shoulders of the inner insertion candidate that physical interference constrains the limit of available insertion space. In real life, as can be see from images shown below and obtained in our laboratory, penny shoulders are rounded. We could not measure the radius of the curvature involved, but it looks like it might very well be sufficient to bollix the theory.
The mechanical limitations on accuracy imposed by the apparatus which displayed a certain amount of slop in actual usage.
The assumption that the ratio of the diameter of a penny to its thickness is that of 3/4 inch to 1/16 of an inch. As will be seen below, this ratio may not hold in real life.

Next, two rulers with finely-engraved scales, and made of humidity and temperature-resistant material, were arranged side-by-side to form a valley between them. This valley being necessary to hold a long column of pennies laid on its side.

Then the two-hundred pennies were removed from their wrappers and arranged in the aforementioned valley in a long stack. It was observed that there was some number of pennies in excess necessary to span the desired 12 inches. This number was determined to be a non-zero positive integer less than two. From this it was calculated that there were 199 pennies in the column. See Figure 2.

Fig 2

For those with a fast Internet connection and who wish to avail themselves
of a more detailed view, a bigger picture (279 Kbytes) is available here.

This result was then audited by removing the pennies from the long stack and attempting to rearrange them in 4 by 5 array of short stacks each containing 10 pennies. It was observed that one such stack was one penny short—confirming the previous result of 199 pennies. See Fig 3. <p> <center> <img src=missing1.jpg alt="Yet more pennies"> <p> <b><u>Fig 3</u></b> </center> During this latter auditing exercise, it was noted by one observer that certain of the younger pennies appeared slimmer than some of the older pennies. Accordingly another experiment was conducted wherein two columns of 10 older pennies (dating mainly from the 1960s) were assembled either side of a similar column, but constituted primarily of younger pennies (dating from 1990 onwards). Subsequently a straight edge was laid across the top and an observer chartered to see if he could see significant daylight passing over the middle stack. See Fig 4. <p> <center> <img src=3_stacks.jpg alt="Pennies again"> <p> <b><u>Fig 4</u></b> </center> From this observation it was concluded that using a non-randomly chosen set of pennies, there could be a variance of as much as somewhere between .3 and .5 of a penny thickness per ten pennies. This was computed to represent a variance of between 3 percent and 5 percent. <p> Further inquiries elucidated the information that 1982 was a transition year. During the course of this year, pennies went from being 100% copper to being made of a zinc core clad in copper. This may account for the difference in penny thicknesses. <H3> Conclusions</H3> It is recommended that this committee declare for Doldrums purposes that a penny is deemed to have a diameter of 3/4 of an inch, and a thickness of 1/16 of an inch. <H3> Potential Avenues for Further Research</H3> It has been brought to our attention that the problem of optimal tesselation of circles within a rectangle is non-trivial. The results of certain theoretical mathematical investigations by members of a distinguished research laboratory located in Murray Hill NJ, are contained in a paper whose abstract (and links for downloading the actual paper) may be found <a href="http://www.combinatorics.org/Volume_3/Abstracts/v3i1r16.html">here</a>. <p> Due to constraints in time and intellectual capacity, the current members of the this research team have no intent of pursuing this further. A quick read of this document made us fear what we might find as a result. </div>  </div> </td> </tr> </table>   <p> </p> <table width="100%" class="footbar"> <tr> <td width="100%" align="center" class="footbar"> <table align=center> <tr> <td class="sl-link" nowrap><a href="/index.shtml">AUE home</a></td> <td> </td> <td class="sl-link" nowrap><a href="#top">Top of this page</a></td> </tr> </table> </td> </tr> </table> </td> </tr> </table> <a name=bottom></a> </body> </html>