Perfect square

The numbers to the right of the decimal point form a perfect magic square: "Each row, column, and diagonal adds to 81.'

I'll accept the assertion as truth, but can someone explain WHY this is true?  I know there are some readers of this blog with quite advanced math skills.

Found in the Futility Closet - the source of many of my best math posts.

A brief history of Arabic numerals and the decimal point

For those interested in this type of material, there is some relevant discussion at this Reddit thread.

"Tell me a truth and you shall have your baby back."

Here is a curious old story that is something like a puzzle: A crocodile stole a baby, ‘in the days when animals could talk,’ and was about to make a dinner of it. The poor mother begged piteously for her child. ‘Tell me one truth,’ said the crocodile, ‘and you shall have your baby again.’ The mother thought it over, and at last said: ‘You will not give it back.’ ‘Is that the truth you mean to tell?’ asked the crocodile. ‘Yes,’ replied the mother. ‘Then by our agreement I keep him,’ added the crocodile; ‘for if you told the truth I am not going to give him back, and if it is a falsehood, then I have also won.’ Said she: ‘No, you are wrong. If I told the truth you are bound by your promise; and, if a falsehood, it is not a falsehood, until after you have given me my child.’ Now, the question is, who won?

– Pennsylvania School Journal, March 1887

Found in the always-delightful Futility Closet.

Mathematical curiosity

The first few powers of 5 share a curious property — their digits can be rearranged to express their value:

25 = 5²
125 = 5^{1 + 2}
625 = 5^{6 – 2}
3125 = (3 + (1 × 2))⁵
15625 = 5⁶ × 1²⁵
78125 = 5⁷ × 1⁸²

It’s conjectured that all powers of 5 have this property. But no one’s proved it yet.

Found in the Futility Closet.

Sorting pennies in the dark

I am recurrently amazed at the wonderful logic and math puzzles posted at Futility Closet.   Here's a recent one:

You’re in a pitch-dark room. On a table before you are 12 pennies. You know that 5 are heads up and 7 are tails up, but you don’t know which are which. By moving and flipping the coins you must produce two piles with an equal number of heads in each pile. How can you do this without seeing the coins?

I was not able to solve this on my own and had to peek at the answer.  Even after seeing the answer, it took me a long time to comprehend why it works.

I'll add some thoughts in the Comments section, but for the answer, go to Futility Closet.

Fiendishly tricky geometry puzzle

Start with a piece of paper 8" x 8", as shown on the left.  Make the two cuts shown.

Then shift the big lower-left piece up and to the left and move the triangle from the upper left corner to the lower right one.

The area that used to be 8x8 = 64 square inches now appears to be 9x7 = 63 square inches.  Where is the fallacy?

I pondered this for way too long, and finally had to cut some physical pieces of paper to solve the paradox.

The puzzle was originally created by or published by the famous Sam Loyd; I found it in the Futility Closet.

101010 is 42 in binary

And 42 is, of course, the "Ultimate Answer to the Ultimate Question of Life, The Universe, and Everything."

It takes Deep Thought 7½ million years to compute and check the answer, which turns out to be 42. Unfortunately, The Ultimate Question itself is unknown.

When asked to produce The Ultimate Question, the computer says that it cannot; however, it can help to design an even more powerful computer (the Earth), that can. The programmers then embark on a further ten-million-year program to discover The Ultimate Question. This new computer will incorporate living beings in the "computational matrix", with the pan-dimensional creators assuming the form of mice. The process is hindered after eight million years by the unexpected arrival on Earth of the Golgafrinchans and then is ruined completely, five minutes before completion, when the Earth is destroyed by the Vogons to make way for a new Hyperspace Bypass...

At the end of the first radio series (and television series, as well as the novel The Restaurant at the End of the Universe) Arthur Dent, having escaped the Earth's destruction, potentially has some of the computational matrix in his brain. He attempts to discover The Ultimate Question by extracting it from his brainwave patterns, as abusively suggested by Marvin the Paranoid Android, when a Scrabble-playing caveman spells out forty two. Arthur pulls random letters from a bag, but only gets the sentence "What do you get if you multiply six by nine"?

Some readers subsequently noticed that 6₁₃ × 9₁₃ = 42₁₃ (using base 13). Douglas Adams later joked about this observation, saying, "I may be a sorry case, but I don't write jokes in base 13."

The number 42 also appears frequently in the work of Lewis Carroll, and some critics have suggested that this was an influence.  Other purported Carroll influences include that Adams named the episodes of the original radio series of The Hitchhiker's Guide to the Galaxy "fits", the word Carroll used to name the chapters of The Hunting of the Snark.

If you're married to or dating a math nerd...

... send them the equations shown and ask them to draw the graph.

Found at Proof Math is Beautiful.

Math teacher fail

Via Reddit.

I need some math help

Kev posted the photo above at Nothing To Do With Arbroath this morning.  It's obviously meant to be humorous, but it should be solvable.

To try to solve it, I assume that the first three - symbols are dashes, not subtraction signs, because she is trying to generate a telephone number.

The cube root of 54,872,000 is 380. 

That gives 1-650-380-....  

In the last figure, if the parenthesized number is 112, the result is of course negative (-12,535), and x7/10 comes to -8774.5.

Does she want it rounded off/up to yield 1-650-380-8775 ??   If so, why did she make it negative?  Or is the symbol before 7/10 actually a negative sign?

I think I need math help, and I don't want to call her.  So I'll blog it...

Addendum:  Reader Steve Blunk has already come up with an answer.  That took all of ?what - 20 minutes?  Here it is:

My take for the last grouping would be

7/10*(9-(11i)^2) =
7/10*(9-(11^2*i^2) =
And since i^2 = -1, the rest follows:
7/10*(9+11^2) =
7/10*130 = 91

but since four digits are required, I'd write it as 0091

So Paula's number is 1-650-380-0091

A geometric "magic square"

Complete "plates" can be assembled from the fragments in any horizontal row, any vertical column, and from either of the diagonals.

Fascinating.  I've not seen one like this before.

Original credit to Lee Sallows, via Futility Closet.

Secret information is encoded in this photo via steganography

It looks like an innocent photo of a chicken that might be emailed by anyone to anyone, or posted on a website with some banal commentary.  In fact, this photo has a message encoded it in.  I have heard of such processes before; the Wolfram Blog explains how the effect is achieved:

The idea of steganography is to hide messages within other information so that no one notices your communications. The word itself comes from a Latin-Greek combination meaning “covered writing”, from earlier physical methods that apparently included tattooing a message on a messenger’s head before letting him grow his hair back to hide it. In the case of digital steganography, it is all done in the math...

Amazingly, it is possible to hide another, larger, full-color picture within this image and get it back again with about a dozen lines of Mathematica code.  The key to the whole process is to use the least significant bit in each color channel of each pixel as a place to hide information...

In a sufficiently complex image, the human eye doesn’t see the loss of information. The fact that one color channel on a particular pixel might or might not be darker by 1/256 than before cannot be noticed…

The technique is explained in detail at the link, and is way over my head, but some of you may understand it.

June 28 - "perfect" day

"This date is the only date each year where both the month and day are different perfect numbers (June 6 being the only date where the month and day are the same perfect number)."

Project Euler

I stumbled across the website for Project Euler, and it sounded intriguing:

"Project Euler is a series of challenging mathematical/computer programming problems that will require more than just mathematical insights to solve. Although mathematics will help you arrive at elegant and efficient methods, the use of a computer and programming skills will be required to solve most problems.

The motivation for starting Project Euler, and its continuation, is to provide a platform for the inquiring mind to delve into unfamiliar areas and learn new concepts in a fun and recreational context."

Then I peeked at some of the math problems:

1. Add all the natural numbers below one thousand that are multiples of 3 or 5.

2. Find the sum of all the even-valued terms in the Fibonacci sequence which do not exceed four million.

4. Find the largest palindrome made from the product of two 3-digit numbers.

7. Find the 10001st prime.

10. Calculate the sum of all the primes below two million.

As Halsey Hall used to say, "Holy cow!"

The Fibonacci sequence of the sunflower

The video above is the impressive "Nature by Numbers" video, which almost everyone has posted this week.  The Nautilus and sunflower analogies are pretty well known, though this was my first encounteer with the mathematics of a dragonfly's wing.

Here's some additional commentary on the sunflower's remarkable structure:

In order to optimize the filling [of the flower head with seeds], it is necessary to choose the most irrational number there is, that is to say, the one the least well approximated by a fraction. This number is exactly the golden mean. The corresponding angle, the golden angle, is 137.5 degrees. (It is obtained by multiplying the non-whole part of the golden mean by 360 degrees and, since one obtains an angle greater than 180 degrees, by taking its complement). With this angle, one obtains the optimal filling, that is, the same spacing between all the seeds.

This angle has to be chosen very precisely: variations of 1/10 of a degree destroy completely the optimization. When the angle is exactly the golden mean, and only this one, two families of spirals (one in each direction) are then visible: their numbers correspond to the numerator and denominator of 2 consecutive numbers in the Fibonacci sequence, which is proved to converge toward the Golden Mean value of 1.6180339... (in the picture we have 21/34, the 7th and 8th terms of the Fibonacci sequence).

Moreover, generally the petals of flowers are formed at the extremity of one of the families of spiral (true, I count 34 for this sunflower). This then is also why the number of petals corresponds on average to a Fibonacci number.

Via the Flickr photostream of lucapost, whence also the illustrative image below.

The words for numbers

This is a most curious observation:

In English, the name of each integer shares a letter with EACH of its neighbors. ONE shares an O with TWO, TWO shares a T with THREE … and so on to infinity.

One

Two

Three

Four

Five

Six

Seven

Eight

Nine

Ten

Eleven

Twelve

.......teens

Twenty-...

Thirty-... etc. etc.

Found at Futility Closet, which posts lots of mathematical curiosities.

Mathematics can be beautiful

A partir d'un carré central de côté 1, on construit un nouveau carré qui s'appuie sur le précédent. Puis on répète la construction, chaque nouveau carré appuie son côté sur l'ensemble des carrés déjà construits. Dans chaque carré, on trace un quart de cercle joignant un sommet au sommet opposé, de sorte que les quarts de cercle soient consécutifs. La courbe obtenue s'appelle la spirale de Fibonacci. 

Photo credit Contra Natura and Labregonet , via Titam.

Math puzzle

The painting above reportedly dates from the 1800s.

It is put on display in Russian most prominent galleries “Tretiakovka” so anyone can check it there by themselves. It shows a scene from a study process in a village school back then. There is a teacher and a bunch of children. Then all attention goes to the blackboard: It has some task for the kids that seem to puzzle them, but that’s just a temporary confusion.  They all look to be ready for the challenge. The name of this art masterpiece is something like “Oral maths test in village school”. It gets what it means - no calcs involved, just mind skills of counting, even without any piece of paper and pencil. So now the question - can any contemporary city school kid do that? I am not sure.

I can't vouch for any of the above, but the Tretyakov Gallery has an immense collection of Russian fine arts, and I see no reason for the painting not to be valid.

But I can vouch for the math puzzle.  You can click to enlarge the painting, but it's probably easier to reproduce it here:

And it CAN be done in one's head without too much difficulty.

Found at English Russia.

Update:  I am recurrently astounded by the breadth of knowledge of TYWKIWDBI readers.  Today Hexmaster identified the painting and the artist - "The painting is called "Counting in their heads". It was made by Nikolay Bogdanov-Belsky in 1895" - and provided a link to the painting's entry in the Russian Wikipedi: The painting on the Russian Wikipedia.

A simple math puzzle

A ball 13 inches in diameter has a 5-inch hole drilled through the center. How deep is the hole?

You can do this one in your head.  The answer is in the comment section.

Found at Futility Closet, where it was credited to Henry Dudeny.

Alice's Adventures in Algebra

There was an fascinating article at New Scientist last month about the influence of Charles Dodgson's mathematics interests on his writing of Alice in Wonderland.

What would Lewis Carroll's Alice's Adventures in Wonderland be without the Cheshire Cat, the trial, the Duchess's baby or the Mad Hatter's tea party? Look at the original story that the author told Alice Liddell and her two sisters one day during a boat trip near Oxford, though, and you'll find that these famous characters and scenes are missing from the text...

...in 1984 Helena Pycior of the University of Wisconsin-Milwaukee had linked the trial of the Knave of Hearts with a Victorian book on algebra... [Victorian] scholars had started routinely using seemingly nonsensical concepts such as imaginary numbers - the square root of a negative number... To Dodgson, though, the new mathematics was absurd, and while he accepted it might be interesting to an advanced mathematician, he believed it would be impossible to teach to an undergraduate.

Outgunned in the specialist press, Dodgson took his mathematics to his fiction. Using a technique familiar from Euclid's proofs, reductio ad absurdum, he picked apart the "semi-logic" of the new abstract mathematics, mocking its weakness by taking these premises to their logical conclusions, with mad results. The outcome is Alice's Adventures in Wonderland. 

The article then goes on to discuss the relationship of the Cheshire cat scene, the tea party, and the other non-original scenes to Dodgson's conservative view of mathematics.

What, then, would remain of Alice's Adventures in Wonderland without these analogies? Nothing but Dodgson's original nursery tale, Alice's Adventures Under Ground, charming but short on characteristic nonsense.

Full details at the link, via NAACAL.  The embedded photos are selections from a wonderful gallery of film stills from Alice in Wonderland ( Clyde Geronomi and Wilfred Jackson, 1951)