A Voice From the Past

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… I know I feel differently in the morning, after seeing the pictures and reading the stories of lives arbitrarily snuffed out and people who had little reduced to people who have nothing at all. But tonight I’m going to bed in a vicious mood, with a stomach so full of contempt for this poisoned republic and its brain-dead citizens that I can taste it in my mouth, like bile.

And the only thing I can think to ask God—if she does exists—is why, just for once, can’t you smite the wicked instead of the innocent?

Posted by billmon at 01:09 AM

(Billmon’s full post can now be found on the Internet Wayback Machine at  https://web.archive.org/web/20051001001007/http://billmon.org/)

This 24 September 2005 post was made during the height (or depths) of the Iraq War during the Bush-Cheney regime.  Also recall the other Bush debacle, Hurricane Katrina, was just the month before, August 23–31, 2005.  We thought we were in the depths of depravity and had no inkling that we were only at the higher rings of hell.

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Another Octagonal Area Problem

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This is a fairly straight-forward problem from the 1999 AIME problems.

“The two squares shown share the same center O and have sides of length 1. The length of AB is 43/99 and the area of octagon ABCDEFGH is m/n where m and n are relatively prime positive integers. Find m + n.”

Answer

See the Another Octagonal Area Problem for solutions.

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Survival of Records

I have written about this a bit in my “Symbolic Algebra Timelines” post as part of the discussion of how Greek mathematics got transmitted to the present.

I had read an article by the renowned literary historian Gilbert Highet some sixty years ago which I never forgot.  It discussed in some detail the miracle of the survival of records from the past.  I tried to find a copy online, but in vain.  So I purchased a used 1962 copy of the now extinct Horizons that contained the article and proceeded to digitize it.  Since the subject of the article is about disappearing writings of the past and how some managed to survive through reproduction, I feel it somewhat apropos that I allow it to see the light of day again.  He is a marvelous writer and covers the subject with fascinating detail.

Even with all the losses Highet records, writings still survived because they were in some sort of hard copy form. By 1962 fragile documents such as newspapers were being moved to microfilm.  But microfilm was often replaced by magnetic tape, which was in turn replaced by a succession of digital media: floppy discs, CDs, DVDs, memory sticks, and so on. Finally, local media is being replaced by files in the “cloud”. All these media are subject to deterioration and loss, or the whims of the custodians. And of course, they all need machines and software to retrieve their information—technology which may no longer exist. Terabytes of moon data are lost on Ampex magnetic tapes for which there are no analog tape readers or even records of the data formats. Even now, some books are being published electronically and not in hard copy.  Some people have referred to this ephemeral situation as the “digital Dark Age” where everything can be lost—often in an instant.  So perhaps we are lucky that records from the past were not digital.

My copy of Highet’s article appears in two forms: a full, but very large, version with all the color figures, Survival of Records (58 MB), and a text-only smaller version, Survival of Records wo figs (700 KB).

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Three Dutchmen Puzzle

Presh Talwalkar presented an interesting puzzle that originated in the Ladies’ Diary of 1739-40, was recast by Henry Dudeney in 1917, and further modified using American money.

“Each of three Dutchmen, named Hendrick, Elas, and Cornelius has a wife. The three wives have names Gurtrün, Katrün, and Anna (but not necessarily matching the husband’s names in that order). All six go to the market to buy hogs.

Each person buys as many hogs as he or she pays dollars for one. (1 hog costs $1, 2 hogs are $2 each, 3 hogs cost $3 each, etc.) In the end, each husband has spent $63 more than his wife. Hendrick buys 23 more hogs than Katrün, and Elas 11 more than Gurtrün. Now, what is the name of each man’s wife?”

See the Three Dutchmen Puzzle for solutions.

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The Umbrella Problem

This is a rather mind-boggling problem from the 1947 Eureka magazine.

“Six men, A, B, C, D, E, F, of negligible honesty, met on a perfectly rough day, each carrying a light inextensible umbrella. Each man brought his own umbrella, and took away—let us say “borrowed”—another’s. The umbrella borrowed by A belonged to the borrower of B’s umbrella. The owner of the umbrella borrowed by C borrowed the umbrella belonging to the borrower of D’s umbrella. If the borrower of E’s umbrella was not the owner of that borrowed by F, who borrowed A’s umbrella?”

Answer.

See the Umbrella Problem for solutions.

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Counting Tanks

A great example of the application of simple math to real-world problems is provided in a recent Numberphile video on YouTube by the British mathematician James Grime.

It is taken from a real story from World War II where the British were trying to estimate the number of tanks the Germans were producing each month.  The spies came up with an estimate of about 1500 tanks per month, whereas the mathematicians estimated the number to be closer to 250 tanks per month.  How the mathematicians did this is explained by Grime.  (I added an explanation of one step that whizzed by in Grime’s presentation and that I didn’t understand at first.)

See Counting Tanks.

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