Tank production

A story is told from World War II, when the Allies were considering a ground invasion of Europe. Critical to their success would be the number of Mark V tanks fielded by the Germans. Intelligence, however, gave contradictory estimates as to the number of tanks that were being produced per month.

It was noticed that captured tanks had serial numbers, which were apparently assigned in order of production. Statisticians came up with a formula which would give an estimate of the total number of tanks produced given a sample of serial numbers. After the war, German production records showed that the estimate was almost exactly correct.

Suppose tanks are numbered 1, 2, 3, 4, 5, ... N, where N is the total number of tanks produced. Given a sample of, say, five serial numbers, how would you estimate the value of N?

Right triangles

Prove that a triangle with sides that can be written in the form of n² + 1, n² - 1 and 2n (where n > 1) is right-angled.

Show, by means of a counterexample, that the converse is false.

[from The Curious Incident of the Dog in the Night-time]

Crossing cylinders

Imagine two cylinders of diameter one with axes intersecting at right angles.

• What is the volume of the space belonging to both cylinders?

• Describe the curve along the intersection you would get by "unrolling" one cylinder like a piece of paper.

(The shape of the intersection is called a "Steinmetz Solid," and Wolfram has spoilers about it.)

Irrationality

In the introduction to his book, Stuart Sutherland offers some sample questions to demonstrate to the reader that rational thinking is sporadic at best. Some are subjective ("Are you a better than average driver?") -- here are the more objective questions:

1. Which is more likely -- that a mother with blue eyes has a daughter with blue eyes or that a daughter with blue eyes has a mother with blue eyes?
2. Are there more words beginning with the letter 'k' than with 'k' as the third letter?
3. Smoking increases the risk of lung cancer by a factor of ten and of fatal heart disease by a factor of two: do more smokers die of lung cancer than of heart disease?
4. Do more people die of strokes than of accidents?
5. Which is more dangerous, cycling or riding the Big Wheel?
6. Consider two maternity hospitals, one averaging forty-five births a day, the other fifteen: in which hospital is it more likely that on any given day 60 per cent of births will be boys?

I wrote down six answers and expect to find the author's correct answers as I read on.

Doctored Photo #8

Ten points.

Ok then, prove *this*

Steve cracked that elementary proof so easily, I thought I'd throw up another interesting one - somewhat related - which I remember from my number theory class. I haven't tried to reconstruct this one yet, but I remember it being somewhat more difficult than the infinitude of primes one.

Prove that for any natural number n, there exist n consecutive composite numbers.

This really did come up in conversation

This really belongs on the lamented Fiat Lux's "e to the pi to the i" board, but since that's no longer with us, I'll use it to check whether we still have any readers here.

---

Not long ago I was in Rhode Island visiting my brother and his family. Somehow the question of whether the number of primes is infinite came up in casual conversation. My sister-in-law was of the opinion that there likely is only a finite number of primes. My brother and I, both being somewhat math nerds, knew that that was contrary to fact.

"Ah-ha, but can you prove it?" Ku asked, a little smugly, knowing our best mathing years were well behind us.

It turns out I was able to recall/reconstruct the quite elementary proof in my head, right there in the kitchen. Anybody care to take a crack at it?

A couple hints:
Assume the contrary and demonstrate a contradiction.
The first step is to construct an integer with certain useful properties.

Sudorku redorx

Another, somewhat drier, question that occurred to me: what is the total number of possible solution grids for a sudoku puzzle? I.e., how many ways are there to consruct a 9x9 grid that has "the sudoku property"?

Su dork u

So this nonsense is popular now. I really don't get it - one grid filled with numbers is about the same as another, to me. With crossword puzzles, at least, the clues can be crafted with varying degrees of cleverness - not to mention 26 > 9.

However, it irks me that there might be peabrains running around getting good at these goofy puzzles and then having a valid claim of mental superiority when I can barely solve the "medium" puzzles in the Wed./Thurs. WaPo. So I've been practicin' some. It's all kind of tedious, but the most tedious part is checking the answer when I'm done.

So here's the nut: In the absence of an answer key, what is the fastest way to check that a su-dork-u grid contains one and only one of each digit in each row, column, and 3x3 box? Put another way: Call each row, column or box an "element"; what is the minimum number of elements that need to be checked for having each digit exactly once in order to be sure that all 27 elements have that property?

Doctored Photo #7

Identify the source of this edited photo and win ten points.

Doctored Photo #6

Here's a chance for a benchwarmer to get on the scoreboard. Five points.

Doctored Photo #5

This one should be within reach of casual news junkies. Ten points to whoever can identify the source.

Find non-prime, win beer cozy

I flipped through Douglas Copeland's book JPod this afternoon and noticed this passage:

Evil Mark stood up in the middle of the afternoon and said "I'm about to hand out sheets listing the 8,363 prime numbers between 10,000 and 100,000. Embedded in this list is one non-prime. First person to find that non-prime number wins my Family Guy promotional sixteen-ounce beer cozy."
In less than five minutes I won.
FUN FACT: any even number can be made by adding together two primes.

The next several pages contained lots of large numbers. The reference to Goldbach's Conjecture might suggest that there is an even number somewhere in the list, but that would seem a comedown from the guy who once encoded a passage in binary.

Any ideas?

Doctored Photo #4

Fifteen points to whoever can identify the source for this retouched photo.

Doctored Photo #3

Here's the most obscure picture so far, worth 20 points.

Doctored Photo #2

This photo is more obscure, and worth ten points if you can locate the unedited original.

Doctored Photo #1

The famous photo below has been doctored to remove the subject of interest. To play, comment with a link to a search that produces the original image in the first page of results. This picture is rated five points, with extra bragging rights if your first attempt finds multiple correct hits on the first page.

Dorks and dragons

So there's this kingdom, and in the kingdom is a lake. In the lake, there's an island, and on the island is a mountain. On the mountain lives a dragon, who's been devouring all the virgins in your village.


Dismayed at this state of affairs, you beard the dragon in his lair. The dragon sneers down at you, and says "I admire your courage, little dude. Tell you what: I propose a contest of wits. On the side of the mountain, there are thirteen poisoned wells, labeled 1-13. All the poisons taste the same - just like regular water. If someone/something drinks a cup of poison from well X, they will die unless they drink another cup of poison from a well numbered greater than X (within a reasonable amount of time). For example, if I drink from well 4, I will die unless I have a cup of poison from well 5 or well 6 or well 7, etc.. So... here's what I propose. Both you and I are rational thinkers. Lets each get a cup of water (without each other seeing) and, then exchange cups. We then do whatever we think is necessary to survive, but we must drink the cup offered to us. How's that for a deal?"

Fortunately for you, you got a look at the layout of the wells on your way to the lair, and you know the dragon has an ace up his sleeve: the dragon can get his water from any of the wells; however, well 13 is impossible for you to reach, being at the very top of the mountain and surrounded by insurmountable cliffs.

Do you take the deal? If so, what is your strategy?

Pirate shootout

Incensed at the imbalanced distribution of doubloons, Darrrian and Edwarrrd challenge Arrrthur to a duel, winner take all. They will stand at the corners of a triangle on the beach and take turns shooting. Each turn will consist of a pirate shooting at anyone he chooses, and the last pirate standing takes the gold. They draw straws to determine shooting order. If Arrrthur always kills his target, Darrrian kills his target 80% of the time, and Edwarrrd kills his target 50% of the time, what strategy should each pirate adopt? Who has the best odds of winning?

Pirate puzzle wankery

Five pirates (Arrrthur, Barrry, Charrrles, Darrrian and Edwarrrd) have stolen 100 doubloons and have agreed on a scheme to decide how to divide them:

Arrrthur must propose a division of the booty, and all the pirates then vote on this proposal.

If it fails to get a strict majority, Arrrthur is executed, Barrry proposes a division, and so on.

Each pirate follows a strategy whose descending priorities are: 1) remaining alive 2) getting maximum gold 3) killing others.

What division does Arrrthur propose?