Steve Kass

Updated June 5, 2012. (Scroll to the end for the update.)

This week’s Numberplay is about a locker room full of, well, lockers – 100 of them, to be exact. The lockers are closed until a janitor (let’s call her Portia) visits the room and opens them all. Then Portia visits the room again and closes every other locker. Later on, she visits the room a third time, opening or closing every third one. And so on, for 100 visits in all.

Head over to the Times, then come back here and enjoy the pictures.

Picture #1. The 100 lockers of the puzzle are represented as a row of tiny squares – red squares for closed doors, green for open. The squares in the top row (all green) show the lockers all open after Portia’s first visit to the locker room.

The second row of squares, alternating green and red, show the lockers after her second visit, and so on.

If you squint at the patterns across the top, I think you can see Portia’s face.

Picture #2. After Portia’s first visit to the locker room, all 100 lockers are open. After her second visit, 50 are open and 50 are closed. Ninety-eight visits later, ten lockers are open. (Just which of the lockers are open in the end is the beautiful result of the puzzle, which you’ll find more easily in the comments to this week’s puzzle than you will by trying to count the tiny squares in the picture here.)

You might wonder – I did anyway – how the number of open lockers changes between Portia’s second and hundredth visit. Here’s a graph.

What a delightfully curious graph! The number of open lockers wanders seemingly aimlessly about the low 50s for a while. It briefly threatens to stay put at 54, then glides down with a few small bumps to a final value of 10.

Intrepid readers can find the Excel spreadsheet I created these pictures from here.

Thanks to Gary Antonick for sharing this great puzzle, which was suggested by Volodymyr Ivanchenko.

Updated June 5, 2012. Back at Numberplay, Gary suggested looking for the number of open lockers sequence at the Online Encyclopedia of Integer Sequences. I didn’t find anything, and I suspect that’s because this puzzle gives different sequences for each initial number of lockers.

So, I put together a SQL Server 2012 script (and posted it at SQL Fiddle here) to generate data for this “number of open lockers” graph for locker rooms with other than 100 lockers.

The graph gets even more interesting when there are more lockers.

From Public Document No. 34 of the Commonwealth of Massachusetts, “Annual Report of the Department of Public Health for the Year ended November 30, 1928,” which can be found here:

It was the last sentence that drew me to this page. Manny was my grandfather, and my mother grew up at the (Lakeville State) Sanatorium. (See RUTGER.)

Once here, though, it was impossible not to puzzle over the several averages in the report that were given to seven-digit precision. For example, 193.9426, the daily average number of patients for the year. In the late 1920s, computing 70,983/366 to seven places wasn’t a snap like it is now. More likely, it required some cranking, punching, or scribbling.

My minimally-informed guess? That the Massachusetts Department of Public Health had one of the early motor-driven calculators, like the Marchand above (center). It’s hard for me to imagine a reason to compute seven digits of precision when fewer would suffice unless it was very little extra work to get the extra digits.

Then again, maybe I’m wrong and the reality was more romantic, and there was a clerk who loved little more than long division. I suppose we’ll never know.

During the 13-year period (1995-2007) for which public data is available, 141 babies were born in Los Angeles County and named Unique. Two were boys.

A great many baby names were more unusual than Unique, including Z (boy, 2007), Q (boy, 2005), Abcde (girl, 2005), Awesome (boy, 2007), Yourhighness (girl, 2007), Queenelizabeth (girl, 2005), Unknown (boy, 2005), and Y (boy, 2007).

Data source: Los Angeles County Department of Public Health [link]

Update (May 5, 2012)

I updated this post last year when I discovered the Social Security Administration’s baby-name database, but it turns out there was more there than I realized. In his Breakfast Experiment™ of this morning, Language Log’s Mark Liberman mentioned this richer source of SSA name data.

As was the case for the SSA data I found last year (in the earlier update, which appears below), there’s no data for baby names that are unique (or almost so). But in recent years, Unique is not unique, so there’s more to report. The chart below summarizes the latest SSA data I have for Unique boys and girls.

Update (June 24, 2011)

The Social Security Administration provides a basic query interface to its national baby-name database. Based on Social Security card applications for births that occurred in the United States, Unique was a common enough girl’s name to make the annual top 1,000 list nine times between 1996 and 2009.

Both the Los Angeles and national figures suggest that as a baby name, Unique is becoming less common.

I’m not sure if I’ll eat all 4.7 parts, but this is my evening snack today, and I can do without any fat against the type.

In today’s New York Times, Ross Douthat opines that

a lack of contraceptive access simply doesn’t seem to be a significant factor in unplanned pregnancy in the United States. When the Alan Guttmacher Institute¹ surveyed more than 10,000 women who had procured abortions in 2000 and 2001, it found that only 12 percent cited problems obtaining birth control as a reason for their pregnancies. A recent Centers for Disease Control and Prevention study of teenage mothers found similar results: Only 13 percent of the teens reported having had trouble getting contraception.

If this makes any sense to you at all, imagine the same argument against requiring seat belts in cars as a way to reduce the number of highway fatalities:

Unavailability of seat belts simply doesn’t seem to be a significant factor in highway fatalities in the United States. When the XYZ Institute examined more than 10,000 highway fatalities, it found that in only 12 percent of the cases was no seat belt available to the deceased. A recent Centers for Accident Prevention study of highway fatalities found similar results: Only 13 percent of passengers killed in traffic accidents were in seats not equipped with a working seat belt.

Or the same argument against just about anything useful that should be readily available:

Unavailability of public waste bins simply doesn’t seem to be a significant factor in the litter problem in the United States. When the XYZ Institute interviewed more than 10,000 people who threw trash on the streets, it found that in only 12 percent of them did so because they couldn’t find a waste bin. A recent Centers for a Cleaner America study of littering found similar results: Only 13 percent of people who littered did so more than 20 yards from a public waste bin.

¹ According to the latest Guttmacher survey, released this month, and based on 2008 data, the rates of pregnancy, abortion, and births among teens in the United States are all at historic lows. According to the report, “A large body of research has shown that the long-term decline in teen pregnancy, birth and abortion rates was driven primarily by improved use of contraception among teens.”

More delicious reasons here.

On the other hand, there’s no reason in the latest salvo against sugar, which popped up offensively in the pages of Nature today.

It’s an opinion piece in Nature’s COMMENT section. It’s not peer-reviewed science, it’s silly, and it’s got this CHART inside that make it look all science-y if you don’t look close.

A chart of what? A chart of who the fuck knows what, that’s what.

The caption says: Global sugar supply … excluding fruit and wine.

The article says: “In many parts of the world, people are consuming an average of more than 500 calories per day from added sugar alone (see ‘The global sugar glut’).”

I say: Global sugar supply and added sugar consumption are two different things. Which is it, supply or consumption? Which is it, all sugar or just added sugar? According to the sideways writing, the chart or its data came from the FAO, but a frenetic half-hour and scores of searches at their web site (without a break for gummies) yielded no answer.

Incidentally, why on a map that appears to provide by-country data is Hawai’i shaded a different color than Alaska and the Lower 48? Would a map of per capita calorie consumption or production — whether of all food or just added food, not only of sugar or just added sugar — look any different? Is 500 calories of who the fuck knows what a lot of who the fuck knows what, or at least enough to justify regulating who the fuck knows what?

Pity the children and the bears.

1937: The budget which President Roosevelt submitted 60 days ago is already becoming obsolete.

1949: A Senate Commerce subcommittee said today the nation’s war-built merchant fleet already is becoming obsolete and “a replacement program of new ship construction may be in order.”

1949: The FCC chief devoted most of the address to the current public discussion over the possibility of television sets now in the hands of the public becoming obsolete in the event new video channels are opened in the so-called “ultrahigh frequencies.”

1958: The premier also pointed out that the Moose Jaw plant was fast becoming obsolete and that there was a possibility in the near future of nuclear power being more universally used.

1960: Old fashioned bomb-throwing assassins are becoming obsolete in an age where computing machines and electronic devices are essential tools of day-to-day existence.

1960: The giant Intercontinental ballistic missiles already may be well on the way to becoming obsolete.

1966: Is your front door becoming obsolete? Studies show that in suburban homes 90 per cent of the traffic is between the garage and a side or  a back door. [In “SURBURBAN BYPASS,” a column of potpourri that incidentally fails to mention spelling.]

1973: New letter sorting equipment being used by the Postal Service at Cincinnati is “becoming obsolete as it is being developed,” according to a report by a House subcommittee staff.

1975: The [Education Commission of the States], which conducts periodic student assessments for the U.S. Office of Education, said experts are suggesting that the written word is becoming obsolete as students lean more on the spoken word.

1984: Building in New Hanover County is becoming so concentrated that our present method of sewage disposal is rapidly becoming obsolete.

1985: Science buildings and laboratories at many universities are becoming obsolete, and their condition threatens to cripple important research in health, engineering and other fields in which the United States now leads, a University of Illinois official told a congressional panel Wednesday.

1986: Rotary-dial telephones are becoming obsolete.

News this week of an Adderall shortage, and this report, which draws into question the widely-held belief that methamphetamines cause brain damage and cognitive impairment, prompt me to rescue an old statistical parody I wrote (and posted on my now-moribund Drew web page) in 2003, a few years before I had this soapbox. The news links above are also well worth visiting.

Cocaine’s brain effects might be long term [“news”]

Insulin’s metabolic effects might be long term [parody]

BOSTON, March 10, 2003 (UPI) — Cocaine and amphetamines
might cause slight mental impairments in abusers that
persist for at least one year after discontinuing the
drugs, research released Monday reveals.

MADISON (NJ), March 16, 2003 — Insulin might cause metabolic
disorders in abusers that persist for at least one year after
discontinuing the drug, research released Monday reveals.

However, experts outside the study said the findings were inconclusive
and pointed out although cocaine has been widely abused for decades,
impaired cognitive function is not seen routinely or even known to exist in
former abusers.

"Overall, the abusers were impaired compared to non-abusers on the function of attention and motor skills," Rosemary Toomey, a psychologist at Harvard
Medical School and the study’s lead investigator, told United Press International.

“Overall, the abusers were impaired compared to non-abusers
on tests of sugar metabolism,” Rosemary Toomey, a psychologist
at Harvard Medical School and the study’s lead investigator,
told United Press International.

Previous studies have yielded inconsistent findings on whether
cocaine abuse led to long-term mental deficits. Some studies found
deficits in attention, concentration, learning and memory six months
after quitting. But a study of former abusers who were now in prison
and had abstained from cocaine for three years found no deficit.

Few studies have looked at the long term effects of insulin
abuse, although doctors and scientists generally believe
the drug is harmful. One study of former abusers who
were now in prison and had abstained from insulin for
three years found a higher than normal death rate.

To help clarify these seemingly conflicting results, Toomey’s team,
in a study funded by the National Institute on Drug Abuse, identified
50 sets of male twins, in which only one had abused cocaine or
amphetamines for at least one year. Amphetamine abusers were
included because the drug is similar to cocaine and could have the
same long-term effects on the body.

To address the lack of careful studies, Toomey’s team, funded by
the National Institute on Drug Abuse, identified 50 sets of male
twins, in which only one had abused insulin for at least one year.

Most of the pairs were identical twins, meaning they share the exact
same genetic pattern. This helps minimize the role biological
differences could play in the findings and gives stronger support to the
mental impairments being due to drug abuse.

Most of the pairs were identical twins, meaning they
share the exact same genetic pattern. This helps minimize
the role genetic differences could play in the findings and gives
stronger support to the impairments being due to insulin abuse.

The abusers, who averaged age 46 and had not used drugs for at least
one year, scored significantly worse on tests of motor skills and
attention, Toomey’s team reports in the March issue of The Archives
of General Psychiatry.

The abusers, who averaged age 46 and had not used
insulin for at least one year, scored significantly worse
on tests of sugar metabolism, Toomey’s team reports in
the March issue of The Archives of General Metabolism.

The tests all were timed, which indicates the abusers have
"a motor slowing, which is consistent with what other investigators
have found in other studies," Toomey said.

The tests all were performed after fasting, which indicates the abusers
have “an impaired metabolism unrelated to diet, which is consistent
with the consensus in the medical community,” Toomey said.

Still, the abusers’ scores were within normal limits and they actually
performed better on one cognitive test, called visual vigilance, which
is an indication of the ability to sustain attention over time. This
indicates the mental impairment is minor, Toomey said. "In real life,
it wouldn’t be a big impact on (the abusers’) day-to-day functioning
but there is a difference between them and their brothers," she said.

The finding is significant, she added, because given that the study subjects
are twins and share the same biological make-up, they would be expected
to have about the same mental status. This implicates the drug abuse
as the cause of the mental impairment.

The finding is significant, she added, because given that the study
subjects are twins and share the same biological make-up, they would
be expected to have about the same metabolic status. This
implicates the drug abuse as the cause of the impairment.

Among the abusers, the mental test scores largely did not vary in
relation to the amount of cocaine or amphetamine used. However,
on a few tests the abusers did score better with more stimulant use.

Among the abusers, poorer test scores were consistently associated
with increased levels of insulin abuse. Among the heaviest abusers,
not one scored better than his non-abusing twin.

"The results seem to me to be inconclusive," Greg Thompson,
a pharmacist at the University of Southern California’s
School of Pharmacy in Los Angeles, told UPI.

“The results seem to me to be conclusive,” Greg Thompson,
a pharmacist at the University of Southern California’s
School of Pharmacy in Los Angeles, told UPI.

This is "because both twins are within a normal range
(and) sometimes the cocaine-abusing twin did better than the
non-abusing twin and sometimes not," Thompson said.

This is “because almost without exception, only the non-abusing
twin is within a normal range (and) the insulin-abusing twin did
worse than the non-abusing twin,” Thompson said.

In addition, cocaine has been abused by millions of people, going
back as far as the 1930s and before, he said. "You’d think you’d be
seeing this as a significant clinical problem and we are not," he said.

In addition, insulin has been abused by millions of people,
and poor sugar metabolism among former insulin abusers
has been reported by physicians going back as far as the 1930s
and before, he said. “This is a significant clinical problem,” he said.

Of more concern to Thompson is the effect stimulants such as Ritalin,
which are used to treat attention deficit disorder, are having on
children. "This would be a much bigger problem I would think if
it’s true stimulants impair cognitive function," he said.

Of more concern to Thompson is the effect daily insulin injections
are having on children. Insulin is commonly prescribed to control
diabetes (frequent urination, weight gain, and fatigue syndrome).
“Many of these children will become former insulin abusers, and
poor sugar metabolism will be a major healthcare issue for
them in the years to come,” he said.

"Before I’d worry about the 46 year-old abuser, I’d want to know about the
3 year old being treated for ADD (attention-deficit disorder)," Thompson said.

“Before I’d worry about the 46 year-old abuser, I’d want to
know about the 3 year old being treated for diabetes,” Thompson said.

It is utterly ridiculous. I have a home in Los Angeles and Palm Springs, and California is famous for high taxes, yet we don’t pay anything close to what I will be paying.

So says Tod Abrams, in a peevish, petulant portrayal of his property tax plight from yesterday’s New York Times.

Toddfoolery, I say. Or, because it can apparently now be spelled with one d, todfoolery. Toddfoolery or todfoolery is foolish nonsense from someone named Tod(d).

This Tod, eerily paralleling last year’s plaintive Todd, is perturbed that “his monthly real estate taxes are poised to surge by more than 400 percent over the next several years.”¹ In 2018, the “unbelievable deal” Tod’s been getting on property taxes will have run its course.

Todfoolery the first. “Unbelievable,” as opposed to “utterly ridiculous,” is no exaggeration.

When he bought a condo at the Orion for $1,575,000 in 2007, Tod’s property taxes were $35.26 [sic] a month. He now pays $373.73/month. In 2018, the estimated tax bill will be $1,629/month (tax-deductible).

None of this was unknown to Tod. The Times discreetly hinted as much by titling its article “421a Tax Exemption: Don’t Say You Didn’t Know.” To be fair, Tod didn’t say he didn’t know about all this. He just said it was utterly ridiculous.

Todfoolery the second. “California is famous for high taxes”

Granted, I’m from New Jersey, where property taxes are high, but $1,629/month doesn’t seem like a lot of property tax for a $1.5M condo. Not that I own a $1.5M condo. Maybe it seems like more to someone from California.

California may be “famous for high taxes,” but any such fame has to do with personal income taxes, not property taxes. Property taxes in California are famous, though. They’re famous for Proposition 13, which keeps them low compared to many other states.

Todfoolery the third and fourth. “don’t pay anything close to what I will be paying”

Q: What does Tod pay in California property tax, anyway, and is it “anything close” to the $1,629/month he won’t be paying in New York in 2018 (because he’s selling his place), which he says it’s not?

A: On his Los Angeles home alone, Tod pays $13,500/year, or $1,125/month in property taxes, and yes, this is “anything close” to $1.629/month.

Todfoolery the fifth. “over 400 percent”

The anticipated New York property tax increase is 336%, which is definitely not “over 400 percent.”²

Todfoolery the sixth. “slashed”

Poor Tod, as we learn from Times reporter Julie Satow, is selling his Orion condo (at left). Tod, “who has relocated to Los Angeles³ to be closer to his son and for work, has already slashed the price three times…”

He’s “slashed the price three times.” The quotation marks indicate that this is a real quote from the Times article, but scare quotes are also in order for “slashed,” or perhaps “three.”

Tod initially listed his place for $1,950,000. After three weeks, he shockingly abandoned hope for a 24 percent-in-four-years profit — despite the recent housing boom — and he (Slash #1) relisted his condo for $1,695,000. Slash #2  saw the listing price plummet from $1,695,000 to $1,645,000, or almost three percent. Three percent! Slash #3 saw the price plunge nearly 4.3 percent further, to $1,575,000.

Shower scene music from Psycho, from mp3skull.com

As much as I love the Times, which is a lot, its Real Estate section, and especially the “let’s feel sorry for people who might not make a six-figure profit on that third residence they bought a very few years ago” articles, especially those that use wrong and misleading numbers in a bid to garner (whose?) sympathies, gets on my nerves. Sometimes it makes me spitting mad.

¹ Where several is seven. This may be a new Times record for the length of time over which something is reported to surge. The previous record appears to have been set in 2006 when the Times reported a “43.1 percent surge over six years.”

² The anticipated increase is $1,629 – $373.73 = $1,255.27, which can easily be seen to be less than four times $373.73.

Note also that if gasoline prices were to fall from $4/gallon to $3/gallon, it would be wrong to report that they had “surged by 75%,” whether the price drop happened in a day or took place over seven years.

The New York Times Manual of Usage and Style ought to explain how to calculate percentages if it doesn’t already. It does, after all, explain the similar problem with the phrase “times smaller.”

³ Tod “relocated” to the Los Angeles home he’s owned continuously since 2002.

See also:
How Do You Arrange Your Cheez-Its? [#2]
How Do You Arrange Your Cheez-Its? [#3]

Figure 1. One serving of Cheez-Its arranged cubically. 27 = 3 × 3 × 3

Figure 2. One serving of Cheez-Its arranged non-cubically. 27 = (9+4) + 1 + (9+4)