[Math Lair] Postcards

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Here's an interesting exercise to test your estimation skills. Before World War I or so, postcards typically were printed to the edge of the postcard, with the picture filling the entire card. During World War I and for a while afterwards, postcard publishers tended to leave a white border around the edge of the card, to save on ink. For example, below you can view an example of a postcard from 1913 (top) and an example of a postcard from the 1920s (bottom):

Postcard from 1913 depicting Zavikon, Thousand Islands
Postcard from the 1920s depicting Zavikon, Thousand Islands, and the smallest international bridge in the world between Ontario, Canada, and New York, United States

Now for the mathematics part. What percentage of ink do you think was saved by leaving a white border instead of printing on the entire area of the card? Before reading on, have a look at the cards above and estimate what percentage of ink would be saved.

Measuring the cards, we find that each card is 5½ inches wide and 3½ inches high. The picture on the bottom card is 5 inches wide and 3 inches high. So, the area covered with ink on the top card is:

3½ × 5½ = 19¼in²

The area covered with ink on the bottom card is:

3 × 5 = 15in²

The savings is

1 − 1519¼ ≅ 22%

So, the savings in ink is 22%; in other words, the border consumes more than 15 of the area of the card. Surprising, isn't it? Our intuition can sometimes deceive us about areas when they are displayed like this. It is a good idea to be alert in the real world for things like this.

If you want to print the cards out and measure for yourself, the images are scaled to 100 pixels per inch. Here's a scale ruler:

[ruler]

See also estimating the circumference of a glass.


If you're interested in more information on the postcards depicted: The top postcard, entitled "Zavikon (The House is in Canada and the Island with the Flagpole in the United States), Thousand Islands, St. Lawrence River," was published by The Valentine & Sons Publishing Co., Ltd., and the card that I scanned was postmarked August 9, 1913. It is numbered 108358 in their series. The bottom postcard, entitled "Zavikon, Thousand Islands (showing the smallest international bridge in the world)", was published by Valentine-Black Co., Ltd., the successor company to Valentine & Sons, and is not postmarked. According to this page, Valentine-Black was active between 1922 and 1933, so sometime in the 1920s seems a reasonable guess for the latter card. The two views appear to be based on the exact same photograph. Note that both islands are actually in Canada; there is a boundary marker on the smaller island but the border actually runs slightly to the south and east.

If you're interested in other pages on this site that are illustrated with vintage postcards, see also "The Mathematics of the Lost Chord" and The Expanding Bridge. If you're looking for more images of old postcards, you can see postcards on Canadian Transport Sourcebook.