# Hjelmslev’s Theorem

On each of these two black lines is a trio of red points marked by the same distances.

The midpoints of segments drawn between corresponding points are collinear.

(Discovered by Danish mathematician Johannes Hjelmslev.)

# Misc

• Liza Minnelli, daughter of Judy Garland, married Jack Haley Jr., son of the Tin Man.
• The Netherlands still sends 20,000 tulip bulbs to Canada each year.
• Every positive integer is a sum of distinct terms in the Fibonacci sequence.
• HIDEOUS and HIDEOUT have no vowel sounds in common.
• “Death is only a larger kind of going abroad.” — Samuel Butler

(Thanks, Colin and Joseph.)

# Day Tripper

A letter from Lewis Carroll to Nature, March 31, 1887:

Having hit upon the following method of mentally computing the day of the week for any given date, I send it you in the hope that it may interest some of your readers. I am not a rapid computer myself, and as I find my average time for doing any such question is about 20 seconds, I have little doubt that a rapid computer would not need 15.

Take the given date in 4 portions, viz. the number of centuries, the number of years over, the month, the day of the month.

Compute the following 4 items, adding each, when found, to the total of the previous items. When an item or total exceeds 7, divide by 7, and keep the remainder only.

The Century-Item. — For Old Style (which ended September 2, 1752) subtract from 18. For New Style (which began September 14) divide by 4, take overplus from 3, multiply remainder by 2. [The Century-Item is the first two digits of the year, so for 1811 take 18.]

The Year-Item. — Add together the number of dozens, the overplus, and the number of 4’s in the overplus.

The Month-Item. — If it begins or ends with a vowel, subtract the number, denoting its place in the year, from 10. This, plus its number of days, gives the item for the following month. The item for January is ‘0’; for February or March (the 3rd month), ‘3’; for December (the 12th month), ’12.’ [So, for clarity, the required final numbers after division by 7 are January, 0; February, 3; March, 3; April, 6; May, 1; June, 4; July, 6; August 2; September, 5; October, 0; November, 3; and December, 5.]

The Day-Item is the day of the month.

The total, thus reached, must be corrected, by deducting ‘1’ (first adding 7, if the total be ‘0’), if the date be January or February in a Leap Year: remembering that every year, divisible by 4, is a Leap Year, excepting only the century-years, in New Style, when the number of centuries is not so divisible (e.g. 1800).

The final result gives the day of the week, ‘0’ meaning Sunday, ‘1’ Monday, and so on.

Examples

1783, September 18

17, divided by 4, leaves ‘1’ over; 1 from 3 gives ‘2’; twice 2 is ‘4.’

83 is 6 dozen and 11, giving 17; plus 2 gives 19, i.e. (dividing by 7) ‘5.’ Total 9, i.e. ‘2.’

The item for August is ‘8 from 10,’ i.e. ‘2’; so, for September, it is ‘2 plus 3,’ i.e. ‘5.’ Total 7, i.e. ‘0,’ which goes out.

1676, February 23

16 from 18 gives ‘2.’

76 is 6 dozen and 4, giving 10; plus 1 gives 11, i.e. ‘4.’ Total ‘6.’

The item for February is ‘3.’ Total 9, i.e. ‘2.’

23 gives ‘2.’ Total ‘4.’

Correction for Leap Year gives ‘3.’ Answer, ‘Wednesday.’

(Via Edward Wakeling, Rediscovered Lewis Carroll Puzzles, 1995.)

# Oh

This is a floodlight photographed at night. What are the segmented stalks that seem to surround it? The phenomenon is seen regularly in photographs and videos; cryptozoologists and students of UFOs call the entities rods.

In 2003 author Robert Todd Carroll consulted entomologist Doug Yanega, who explained that they’re flying insects (in this case moths).

“Essentially what you see is several wingbeat cycles of the insect on each frame of the video, creating the illusion of a ‘rod’ with bulges along its length,” Yanega wrote. “The blurred body of the insect as it moves forward forms the ‘rod,’ and the oscillation of the wings up and down form the bulges.”

“Some hilarious photographs of ‘rods’ have been posted on the Internet,” Carroll noted. “My favorite is ‘the swallow chases a rod’ which looks just like a bird going after an insect.”

# In a Word

aegritude
n. an instance of sickness

Utah senator Jake Garn got so comprehensively ill on the space shuttle Discovery in 1985 that he’s remembered in the Garn scale, an informal measure of space sickness. Astronaut Robert Stevenson recalled:

Jake Garn was sick, was pretty sick. I don’t know whether we should tell stories like that. But anyway, Jake Garn, he has made a mark in the Astronaut Corps because he represents the maximum level of space sickness that anyone can ever attain, and so the mark of being totally sick and totally incompetent is one Garn. Most guys will get maybe to a tenth Garn if that high. And within the Astronaut Corps, he forever will be remembered by that.

Garn said, “I’ve been very proud of the fact that they named something after me after all these years, even if it was unofficial.”

# Inventory

From Lee Sallows:

(Thanks, Lee!)

# Fancy That

In 2005 Yale psychologists Deena Skolnick and Paul Bloom asked children and adults about the beliefs of fictional characters regarding other characters — both those that exist in the same world, such as Batman and Robin, and those that inhabit different worlds, such as Batman and SpongeBob SquarePants.

They found that while both adults and young children distinguish these two types of relationships, young children “often claim that Batman thinks that Robin is make-believe.”

“This is a surprising result; it seems unlikely that children really believe that Batman thinks Robin is not real,” they wrote. “If they did, they should find stories with these characters incomprehensible.”

One possible explanation is that young children can find it hard to take a character’s perspective, and so might have been answering from their own point of view rather than Batman’s. In a second study, kids acknowledged that characters from the same world can act on each other.

But this is a complex topic even for grownups. “James Bond inhabits a world quite similar to our own, and so his beliefs should resemble those of a real person. Like us, he should think Cinderella is make-believe. On the other hand, Cinderella inhabits a world that is sufficiently dissimilar to our own that its inhabitants should not share many of our beliefs. Our intuition, then, is that Cinderella should not believe that James Bond is make-believe; she should have no views about him at all.”

(Deena Skolnick and Paul Bloom, “What Does Batman Think About Spongebob? Children’s Understanding of the Fantasy/Fantasy Distinction,” Cognition 101:1 [2006], B9-B18. See Author!, Truth and Fiction, and Split Decision.)

Does the green dot above flash before, as, or after the red dot reaches it? Most people say after, but in fact the flash occurs before the red dot arrives (below). This anomaly is known as the flash-lag effect, and its cause is unclear. Possibly it’s a sign that the visual system extrapolates the position of a moving object more readily than that of an unpredictably flashing one.