Innergrams and what they can tell us about word favoritism

So I was playing Word², and I had a rack of A R T T U _ _ and a desire to build down from the word LEND. Reflexively, I spelled DART and was about to move on when I paused and asked myself "Why didn't I make DRAT?". DRAT is a perfectly fine word, and unlike DART, I don't recall ever playing it before. After building off the end of LEND, I was planning to build another word off the end of DART to extend my slaloming, weaving string of words off toward the horizon. This style of wordcrafting is not atypical, so many people must have previously encountered a similar choice between two words that start and end with the same letters. I began to wonder how they chose.

Fortunately WordSquared has a new feature that allowed me to find out. By clicking on a word on the board, you can pull up a pop-up box containing a little information about the word including definitions, who has recently played the word, and how many times it has been played (since statistics have been kept... about a month ago as of late-March).

DART had been played 4789 times.

DRAT had been played 1723 times.

So it was not just me.

I decided to study this a little more. I compiled a long list of anagrams that share the same first and last letters (e.g., FORTH and FROTH, SEAHORSE and SEASHORE). Since it is only the inner letters that are scrambled, I decided to call them "innergrams".

The table below shows the resulting innergram pairs with each word's respective word count (as taken from Word² statistics pages like this one.) The words are sorted so the more frequently used one is always in the first column. The fifth column shows the ratio of the two word counts.

Since I wanted to identify data that was not strong enough to draw conclusions from, I used formal hypothesis testing. The chi-square goodness of fit test I used is described in detail here. The essence of it is that the more data you have and the farther the ratio of word counts is from 1, the stronger the evidence is that one word is preferentially being used over the other. The chi-square parameter (in column 6) measures how strong this evidence is. I've sorted the table by increasing evidence strength.

Admittedly, there are lots of situations where one of these words would be favored over another for in-game reasons (like, CRAVE was already on the board and CRAVEN was made by just adding an N, or maybe a triple-letter score square made CAVERN a higher scoring choice). Averaged over many instances, some of these effects should cancel out.

The first three rows have such a small chi-square value that it's pretty certain that people are not (on average) favoring one of these words over another. (Maybe for every person who makes CRAVEN by adding an N to CRAVE, there is someone else making CAVERN by adding an N to CAVER.) The gray rows are weakly supported. The rest of the rows have a big enough chi-square parameter that we can say with greater than 95% certainty that Word² players favor the first word over the second word. In the last column of the table, I suggest reasons why.



Essentially, this is a listing of possible word blind spots. DART is a far more popular choice than DRAT, and unlike many of the examples on this list, this asymmetry cannot be explained by ART being a more frequently available hook than RAT. (RAT has been played 19,000 times and ART only 12,000 times.)

I have highlighted with orange the rows where there is strong evidence that the second word is a blind spot word. The green rows indicate that I suspect a blind spot exists, but other explanations could also account for the imbalance.

All innergrams are potentially useful tools for Bananagrams players since the ability to most quickly rearrange your grid can be the difference between winning and losing a game. Blind spot words are just the innergrams that you are most likely to not have immediately at hand... until now!

Blind spot words:
blot, causal, citric, coral, drat, garb, labile, prefect, reserve, rogue, slat, sloe, snag, stanch.

Other possible blind spot words:
brunt, clod, froth, gird, median, recuse, sidle, spilt

---

In the interest of completeness, below are the innergram pairs that I left out of the table because their word counts were too low. (No count exceeded 8.)

scalar	sacral
martial	marital
converse	conserve
eternity	entirety
preserve	perverse
coagulate	catalogue
seashore	seahorse
parental	paternal
compliant	complaint
observe	obverse
reprise	respire
metronome	monotreme
percept	precept

The last two pairs had word counts of zero. Build one of these words in Word² and you may be the first!

On the effect of body orientation on anagramming speed

When we are standing up, gravity pulls blood down which is detected by "baroreceptors" in the circulatory system. The circulatory system compensates by increasing the heard rate, so that blood pressure in the upper body is maintained. When the baroreceptors register such a blood pressure drop, the locus coeruleus neurons become more active. The locus coeruleus responds to stress by increasing release of norepinephrine (a.k.a., noradrenaline). This has a number of effects, including increasing heart rate and blood flow and boosting motivation, thinking, and alertness. But researchers have found that increases in norepinephrine coincide with a slight slowing of certain kinds of thinking.

The thesis is that "insight problems", the kind where your brain kind of wanders and does background processing, and then suddenly presents a solution in an "A-ha!" moment, happen faster when you have lower norepinephrine levels. A simple "A-ha"-type problem to test, and one which is very popular in psychological research, is anagramming.

In one experiment [M.P. Walker, C. Liston, J.A. Hobson, R. Stickgold, "Cognitive flexibility across the sleep-wake cycle: REM-sleep enhancement of anagram problem solving", Cognitive Brain Research, Volume 14, p.317-324 (2002)], researchers woke people up at different stages of sleep and asked them to solve anagrams. They found that subjects who had just been in REM sleep were better at anagramming than subjects who had been in some other sleep stage. And REM sleep has a lower locus coeruleus activity level.

In another experiment, [D. M. Lipnicki and D. G. Byrne, Cognitive Brain Research, Volume 24, p.719-722 (2005)], researchers had subjects solve 5-letter anagrams (like unscrambling DEFSU to get FUSED) and perform arithmetic problems (73 - 58 + 19 - 26 = ?), while standing up and then while lying down. When standing up, average anagramming time was 29.4s +/- 6.3s. When lying down, average anagramming time was 26.3s +/- 5.4s. The difference in anagramming times (3.1 seconds) is smaller than the standard deviation of the data, but the statistical analysis suggests that the error bars are not that important, allowing us to conclude that the average person in the study anagrammed 10% faster while lying down.

A similar test found that standing is correlated with people solving simple arithmetic problems a few percent faster, but it's such a small difference that it doesn't seem statistically significant.

Summarizing: When you stand up, your body tends to become more alert and less relaxed, and you may anagram slightly slower as a consequence.

Caveats: This study was performed on randomly selected subjects with no proclivity for word games. When given 32 5-letter anagrams to solve, and 45 seconds to solve each in, the average number of solutions was 8 or 9 (whether sitting or standing). A typical Bananagrams player would do significantly better, I would be willing to bet.

I wonder if the cause and effect might flow the other way: if lower blood pressure allows one to anagram faster, can focusing on anagramming cause blood pressure to be temporarily lowered? Or will it just cause one to lie down?

Anagramming is only one of the skills used in playing word games, of course. The kind of concentration necessary to play Bananagrams fast most likely requires the mind to not be in a relaxed state. (And what I'd really like to see is a study on playing Bananagrams itself). Still, whenever the professional Bananagrams league launches, one of the first drugs they should consider regulating is beta-andrenoceptor antagonist propranolol.

---

Further reading:

D. M. Lipnicki and D. G. Byrne, "Thinking on your back: Solving anagrams faster when supine than when standing", Cognitive Brain Research, Volume 24, p.719-722 (2005).

If you have access to it through your library, you can get the paper from here.

History of anagrams: Anagramming as recreation and social skill

Anagrams were very popular during the 17th century, and particularly in France. King Louis XIII used anagrams as a recreational form of intellectual stimulation and source of entertainment for himself and members of his court. He formed some on his own and hired a Royal Anagrammatist whose sole task was to devise anagrams. A lot of these anagrams were based on the names of people, and particularly, on the name of the king with the theme being how great he was.

The book "Of Anagrams" by Henry Benjamin Wheatley (downloadable for free from its Google Books entry) says that "Anagrams were used by fine gentlemen to add pungency to their conversation". It cites Henry Peacham's "Compleat Gentleman" (written in 1634) which basically says that among the conversational skills of a gentleman ("merry tales, wittie questions and answers", "ingenious epigrams") should be the spontaneous spouting of anagrams. He gives the example of someone who was going to say "I must goe buy a dagger" but decided to switch around the letters and say "I must goe dye a beggar". [It's not strictly an anagram though; it's actually a rearrangement of sounds. A more precise description might be "anaphone".]

In the 19th century, it was popular to anagram the names of famous people (celebrities, I suppose). Lewis Carroll produced many such anagrams, including Florence Nightingale ("Flit on, cheering angel."), Disraeli ("I lead, sir."), and a politician of the time named William Ewart Gladstone ("Wild agitator means well!"). The popularity of anagramming at the time led to the Victorian game of Anagrams (previously described here and which now is sold under the name "Snatch") which was played by Alfred Butts as a child. His first attempts at inventing a game drew inspiration from the idea of anagramming as the basis of a game, to which he added a letter distribution like that of the English language. After a few more iterations, he converged on the game of Scrabble, which led to Speed Scrabble, which led to Bananagrams.

History of anagrams: Anagrams as encryption

Galileo was a professor in Padua, Italy, studying physics and mathematics, when he heard about the sudden development of telescopes in Holland. Guessing at how they worked, he created his own in 1609 and began promoting it for the monitoring of ships by merchants.
By the fall, he began using it for astronomical observations. He designed a series of successively better telescopes, giving him the ability to discern smaller details of the night sky than anyone before him. This opened up a gold rush of scientific exploration. Galileo was able to make astronomical discoveries very fast, but such discoveries could sometimes take months or years to verify, which left him with a dilemma: If he announced his discovery immediately, he could eventually turn out to have made a mistake; if he waited, one of his competitors might find the same thing and get the credit. Galileo's solution (and possibly one already in use by other scientists) was to write a short description of his findings, rearrange the letters, and distribute the coded version. In that way, he could reveal his conclusions at any time in the future.

In 1610, he sent letters to his fellow scientists, containing the following string of letters:

smaismrmilmepoetaleumibunenugttauiras

Fellow astronomer, Johannes Kepler anagrammed very hard and came up with this solution:

Salve umbistineum geminatum Martia proles.

which translates to "Be greeted, double knob, offspring of Mars", and which Kepler interpreted to mean that Galileo had discovered that Mars has two moons, something that Kepler had predicted. But it turned out that Kepler's anagram was not the one that Galileo had had in mind. Allegedy the Holy Roman Emperor became interested in Galilelo's finding, and so Galileo finally revealed the actual original sentence to be

Altissimum planetam tergeminum observavi.

which translates roughly to "I have observed the most distant of planets (Saturn) to have a triple form" (where U and V are treated as interchangeable in Latin). Galileo thought that he had observed two moons orbiting Saturn. He was wrong, as Huygens showed in 1656 that what Galileo had seen was actually a ring around Saturn. Huygens wrote an anagram about this, too.

In December of 1610, Galileo made an even bigger discovery which he transmitted as a true anagram:

Haec immatura a me iam frustra leguntur oy.

which, translated, means roughly "This was already tried by me in vain prematurely". Kepler pleaded with him to reveal what he had found, and so in January, Galileo replied with the unanagrammed sentence:

Cynthiae figuras aemulatur Mater Amorum.

Translation: "The Mother of Loves [Venus] imitates the figures of Cynthia [the Moon]." by which Galileo meant that Venus cycles through phases, just like the Moon. He had observed that Venus is not a light source - it simply reflects light from the Sun. And from the way it waxed and waned like the Moon, he could tell that Venus was orbiting the Sun. At a time when many held that everything revolved around the Earth, this was an amazing result.

In the 1670s, Robert Hooke was studying the physics of springs. He found that when you stretch a spring, the force that it pulls back with is proportional to the distance that you pull it. This came to be known as Hooke's Law. Hooke encrypted this in a popular way among scientists of the time: He simply alphabetized the letters and got "ceiiinosssttuv". The unscrambled version is "Ut tensio sic vis." which he revealed a few years later to mean "As the extension, so the force.".

Hooke held on to another discovery much longer. In 1671, he announced to the Royal Society of London (a group of scientists, dedicated to advancing scientific knowledge) that he had figured out what the optimal shape for the arch of a bridge was. Four years later, he published the encoded form of his conclusion:
The unanagrammed form was not revealed until after he died in 1703(!). It read "As hangs a flexible cable, so inverted, stand the touching pieces of an arch." which means that if you hang a chain between two poles, you get a special curve called a catenary, and if you turn it upside down, this is the best shape for supporting a bridge.

I am guessing that the practice of anagramming conclusions in this way gradually disappeared as more formal publication methods were developed (the earliest probably being the journals published by the Royal Society of London starting in 1665).

In a way, this kind of encryption was not so different from the seeking of metaphysical truth through anagramming (as covered in a previous post), except that the scientists were deliberately putting the distilled form of the truth on the other end of the anagram and letting people search for it.


For more background on Galileo and his anagrams, you can read http://www.mathpages.com/home/kmath151.htm.

For more about Hooke's law and his caternary findings, see: http://www.lindahall.org/events_exhib/exhibit/exhibits/civil/design.shtml.

And for a fictionalized account of 17th century science and the development of the Royal Society (among other things), try Neal Stephenson's Baroque Cycle.

History of anagrams: Anagrams as prophecy

One half of the Bananagrams family tree was sketched out in this post which basically covered the inheritance of traits from crossword puzzles by way of Scrabble. But the inventor of Scrabble had another influence beyond word games that arranged words in intersecting columns and rows: An emphasis on anagramming. So, I've dug into the historical roots of anagrams. Here is some of what I have found:

While evidence from the time of the Greeks is spotty, it is thought that Pythagoras and his Pythagoreans may have been rearranging the letters of words in the 6th century B.C., as they believed that anagramming someone's name could reveal information about their destiny. Later, in 332 B.C., Alexander the Great was in the midst of a six-month siege of the city of Tyre, when one night he had a nightmare about a satyr trying to catch him. He asked his soothsayers about it, and they noted that in Greek the word for "satyr" anagrams to "Tyre is thine.". The next day, Alexander finally captured Tyre.

From a very insightful book called The Puzzle Instinct: The Meaning of Puzzles in Human Life by Marcel Danesi:

Soothsayer status was, in fact, often attained by those who claimed to possess knowledge of anagrams. In the third century B.C., for instance, the Greek poet and prophet Lycophron made a profession of devising anagrams of the names of the members of the Hellenistic king Ptolemy II's court in Egypt, as a basis for divining each persons' character and destiny. For this he became widely known and sought out as a soothsayer.

After the time of the Romans, there is little evidence of anagram use until its resurgence in the Middle Ages. As most educated people were associated with the Church, most anagrams from that era relate to religion and are in Latin. The most popular example is this imagined exchange between Pontius Pilate and Jesus:

Pilate: Quid est veritas? ("What is truth?")

Jesus: Est vir qui adest. ("It is the man before you.")

I have to admit, it would be really cool to answer somebody with an anagram of their own question.

The Kabbalists were mystics from the Middle Ages for whom anangramming held particular importance. To understand their interest in manipulating letters and numbers, it helps to know a bit about the ancient Hebrew language.

Ancient Hebrew did not have a separate system for numbers, so the letters of the Hebrew alphabet were used instead. This led to numerology (adding up the numerical values of letters in a word, and regarding the sum as significant to the word). Ancient Hebrew also did not have a conventional alphabet; it had an "abjad" which is a writing system where vowels are not written, but inferred by the reader. This made anagramming much easier. (Just imagine how much easier it would be to play Bananagrams if you could use "NRG" for "energy", "GRN" for "green", and "RNG" for "ring"!)

For the Kabbalists, the letters of the Hebrew alphabet held special powers. They developed a wide variety of rule-based systems for transforming one word into another. Some, like Atbash, were simple substitution cyphers (A->Z, B->Y, C->X,...). Temurah (the Hebrew word for "permutation") was the Kabbalistic art of anagramming. The Kabbalists believed that through such manipulation of names and verses in the Torah, they could uncover fundamental knowledge about the universe. They wore anagrams on amulets which they believed protected them from evil (a practice common among many others in the ancient world)

My guess is that if they had had the opportunity to play, the Kabbalists would have been Bananagrams maniacs.

More anagramming history, coming in a future post.

Rearranging Bananagrams

BANANAGRAMS can be anagrammed into a lot of phrases. My favorite is the headline-like sentence "Agar Nabs Man".

But the most disturbing is not an anagram at all. Just add a space, and you get "Ban anagrams!"! Banning anagrams would be the position of the bad banana (with the greasy black peel).