Dues Process

A curious puzzle by Dartmouth mathematician Peter Winkler: You’ve just joined the Coin Flippers of America, and fittingly the amount of your dues will be decided by chance. You’ll name a head-tail sequence of length 5, and then a coin will be flipped until that sequence appears in five consecutive flips. Your dues will be the total number of flips in U.S. dollars; for instance, if you choose HHHHH and it takes 36 flips to produce a run of five heads, then your annual dues will be $36. What sequence should you pick?

At first it seems that it shouldn’t matter — any fixed sequence should have the probability (1/2)5, or 1/32. But “Not so fast,” Winkler writes. “Overlapping causes problems.” It is true that in an infinite sequence of random flips, the average distance between one occurrence and the next of any fixed sequence is 1/32. But if you choose HHHHH (for example), one occurrence of this outcome gives a huge head start to the next — if the next flip is a tail, then you’re starting over cleanly, but if it’s a head then you’ve already produced the next occurrence.

“If X is the average time needed to get HHHHH starting fresh, the average of 1 + X and 1 is 32,” Winkler writes. “Solving for X yields a startlingly high 62 flips.” To get your expected dues down to $32, you need to pick a sequence where this “head start” effect doesn’t obtain. There are 10 such sequences; one is HHHTT.

(Peter Winkler, “Coin Flipping,” Communications of the ACM 56:11 [November 2013], 120.)

| Science & Math

Succinct

An observation by Oxford University mathematician Nick Trefethen:

A student leaves university in America with a transcript full of information. Even with grade inflation, there are thirty marks of A or A- or B+ or B to look at, each one attached to a different course like Advanced Calculus or 20th Century Philosophy or Introduction to Economics. Grade-point averages are constructed from these transcripts and reported to three digits of accuracy.

An Oxford graduate finishes with no transcript, just a degree result which may be a First, a II.1, a II.2, a Third, a Pass, or a Fail. Failures are more or less nonexistent, and the numbers last year [2000] for the other degrees were 691, 1925, 374, 39, and 3, respectively. The corresponding probabilities are 23%, 63%, 12%, 1%, and 0.1%.

If you add up these probabilities times their base 2 logarithms, all times minus one, you find out how much information there is in an Oxford degree. The result is: 1.37 bits of information.

(From Trefethen’s Index Cards, 2011.)

| Science & Math

A Confusing Census

There are 12 people in a room. Some always tell the truth, and the rest always lie.

#1 says, “None of us is honest.”
#2 says, “There is not more than 1 honest person here.”
#3 says, “There are not more than 2 honest people here.”
#4 says, “There are not more than 3 honest people here.”
#5 says, “There are not more than 4 honest people here.”
#6 says, “There are not more than 5 honest people here.”
#7 says, “There are not more than 6 honest people here.”
#8 says, “There are not more than 7 honest people here.”
#9 says, “There are not more than 8 honest people here.”
#10 says, “There are not more than 9 honest people here.”
#11 says, “There are not more than 10 honest people here.”
#12 says, “There are not more than 11 honest people here.”

How many honest people are in the room?

Click for Answer
| Puzzles

You Are There

Prague artist Robert Barta’s installation Crossing Half a Million Stars consists of 500,000 ball bearings covering the floor of a room.

The visitors themselves create a spontaneous performance as they try to make their way across it.

| Art

“Pussy and the Mouse”

https://books.google.com/books?id=FS8PAAAAQAAJ&pg=PA30

A puzzle by Henry Dudeney:

‘There’s a mouse in one of these barrels,’ said the dog.

‘Which barrel?’ asked the cat.

‘Why, the five-hundredth barrel.’

‘What do you mean by the five-hundredth? There are only five barrels in all.’

‘It’s the five-hundredth if you count backwards and forwards in this way.’

And the dog explained that you count like this:

1     2     3     4     5
9     8     7     6
     10    11    12    13

So that the seventh barrel would be the one marked 3 and the twelfth barrel the one numbered 4.

‘That will take some time,’ said the cat, and she began a laborious count. Several times she made a slip, and had to begin again.

‘Rats!’ exclaimed the dog. ‘Hurry up or you will be too late!’

‘Confound you! You’ve put me out again, and I must make a fresh start.’

Meanwhile the mouse, overhearing the conversation, was working madly at enlarging a hole, and just succeeded in escaping as the cat leapt into the correct barrel.

‘I knew you would lose it,’ said the dog. ‘Your education has been sadly neglected. A certain amount of arithmetic is necessary to every cat, as it is to every dog. Bless me! Even some snakes are adders!’

Now, which was the five-hundredth barrel? Can you find a quick way of arriving at the answer without making the actual count?

Click for Answer
| Puzzles

Oh Well

https://commons.wikimedia.org/wiki/File:Sam_Hughes_holding_the_McAdam_shovel.jpg

In 1913 Canadian politician Sam Hughes proposed the MacAdam Shield Shovel, a spade that could double as a protective shield in the trenches — it was provided with a hole through which a soldier could survey the enemy.

Some 20,000 had been manufactured before it was discovered that the blade was not remotely bulletproof, and it wasn’t much use as a shovel … because there was a hole in it.

| Technology

The Frog Battery

https://en.wikipedia.org/wiki/File:Matteucci%27s_frog_battery_trimmed2.jpg

Early experimenters in electricity sometimes dealt in frogs’ thighs. Dissecting a frog creates an “injury potential” in its muscles, which can then be arranged in series to produce a kind of biological battery. Carlo Matteucci strung together 12 to 14 half-thighs to make a “frog battery” strong enough to decompose potassium iodide; he was able to induce some effect even with living frogs.

Matteucci did similar work with eel, pigeon, and rabbit batteries. In 1803 Giovanni Aldini used a galvanoscope made of frogs to detect current in a circuit that ran from an ox’s tongue to its ear through Aldini’s own body. The mechanisms underlying these results weren’t always clearly understood, but they formed important early strides in bioelectrochemistry.

| Science & Math

Maps and Symbols

https://commons.wikimedia.org/wiki/File:BSicon_FOREST.svg
Image: Wikimedia Commons

This map shows mountain symbols above a river symbol. Suppose that, in the part of the world that the map represents, there really are mountains in the location that the map indicates. But suppose that there are also mountains on the other side of the river — where no mountains are indicated on the map. Would we say that the map is inaccurate?

People tend to say yes — in general, if a marker appears on a map, then we tend to think that the absence of the marker reflects an absence of that feature from the corresponding location.

But this is very different from linguistic representation. “After all,” writes Rutgers philosopher Ben Bronner, “if I say that there are mountains north of the river, the accuracy of my assertion doesn’t depend on whether there are mountains south of the river.”

And suppose I drew a map on which some national capitals were indicated, but not all. We wouldn’t take this to mean that the absent capitals don’t exist, merely that the map is incomplete. So how can we make sense of the intuition?

(Ben Bronner, “Maps and Absent Symbols,” Australasian Journal of Philosophy 93:1 [2015], pp. 43-59.)

| Oddities

Vertigo

https://commons.wikimedia.org/wiki/File:Bozouls_(12)_Le_Trou_de_Bozouls_-_05.jpg
Image: Wikimedia Commons

The town of Bozouls in the south of France sits at the edge of a horseshoe-shaped canyon 300 feet deep, the product of 2 million years of erosion of the region’s limestone plateau by rivers and glaciers.

Because the outcrop at the center of the horseshoe is accessible only from the south, it makes an ideally defensible position, and a castle was built there in the 9th century, of which only ruins remain. In medieval times guards in towers monitored the approach 24 hours a day.

One historic building still survives: The 12th-century St. Faustus church sits right on the edge of the cliff, looking over the river.

| Oddities