This relationship can be utilized as a trick by writing 12345679 and asking a person to select his favorite digit. Mentally multiply the digit he selected by 9, then write the result under the number above. Then say that inasmuch as he is fond of that digit he shall have plenty of it. Multiply the two numbers together and the digit he selected will result. Thus suppose 4 was selected; multiply 12345679 by 36, resulting in 444444444.

— Albert Beiler, Recreations in the Theory of Numbers, 1964

(I don’t know who came up with this — there’s a whole series of memes on this theme.)

If f(z) is a cubic polynomial with complex coefficients, and if the roots of f are three distinct non-collinear points A, B, and C in the complex plane, then the roots of the derivative f′ are the foci of the unique ellipse inscribed in triangle ABC and tangent to the sides at their midpoints.

The theorem is named for Morris Marden, but it had been proven about a century earlier by Jörg Siebeck.

(Dan Kalman, “The Most Marvelous Theorem in Mathematics,” Math Horizons 15:4 [April 2008], 16-17.)

There are only five integer-sided triangles in which the area equals the perimeter:

(5, 12, 13) (6, 8, 10) (6, 25, 29) (7, 15, 20) (9, 10, 17)

Only two of them are right triangles.

(Lee Markowitz, “Area = Perimeter,” Mathematics Teacher 74:3 [March 1981], 222-223.)

In trying to work out the trajectories of the planets, Johannes Kepler had to do reams of monstrous calculations by hand. In the manuscript pages for his revolutionary Astronomia nova of 1609, he concludes 15 folio pages of computations by writing:

“If thou art bored with this wearisome method of calculation, take pity on me, who had to go through with at least seventy repetitions of it, at a very great loss of time.”

The Book of Common Prayer includes a Table of Kindred and Affinity that lists prohibited degrees of marriage in the Church of England. For example, a man may not marry his daughter’s son’s wife, and a woman may not marry her husband’s mother’s father. In this case, the two proscriptions correspond — they describe the same relationship “from both sides,” so this union is prohibited to both parties in the relationship. But is this always the case? Is each union that’s denied to a man also denied to the woman? (The table lists only heterosexual unions.) It’s not immediately clear; 25 prohibited degrees are listed for each sex, and our language makes it hard to “reverse” the description of a relationship mentally.

In 1989 Manchester Polytechnic mathematician M.D. Stern worked out a notation that makes this easy. Use 1 to denote a male and 0 a female, and use this code to denote relationships between individuals:

00 spouse
01 parent
10 child
11 sibling

Now, to show the relationship between one person and another, write one digit for the first person followed by a sequence of three more digits — two to represent the relationship and one to represent the sex of the second person. So, taking the example above, a man’s daughter’s son’s wife would be denoted:

1 100 101 000

To interpret the same relationship from the woman’s point of view, we just reverse the order of the digits:

0 001 010 011

He is her husband’s mother’s father.

Applying this to the prohibited degrees in the table, Stern found that every prohibition for a man corresponds to an inverse prohibition for a woman — there are no prospective marriages that would be prohibited to one party but not the other.

(M.D. Stern, “A Notational Device for Analysing Relationships,” Mathematical Gazette 73:463 [March 1989], 37-40.)

Inspired by James McKee’s Trinity Hall prime, physics researcher Gilles Esposito-Farese (of the self-descriptive pangram) has worked out that this 2,258-digit prime number:

53084469288402965415857953902888840109250315594591772197851275320
59910735745658243457138160802170063601085186072703319516241231606
86858731799078163479147444957979157038109676507221794134810159187
99946828292780972255445123198357952762990102770564813212521111380
68349356142222060588948481473772481328151681322128358047354205784
70540373426870045719371801557904317944239925854314103160489406736
99060594293236420918525997085793619098456109204165164418661475892
01109662597018777150106134376006906212249382614768188594613749419
81773446881694503562852062669737115544391406458301430146238093071
02894114746014041621168186006763973309046545159248106457826024237
26295585692705999572335711556642484343647905815411003310539537633
41950800883057333667657184487306007957156203546941504909814030908
34965188540308705963440466656812927154037805823279990648845960204
63331562527077555356154644447566217362506777244670808476000607805
03811498534406491478259767678703610171309197408223291080531370612
62650405840051780819121599354652788179742394248611555080762986718
96826790066089904275315943211982421764246751417927128802586925712
27099955857542532516878558368313422620050604202219808465512659996
23064148740328947837353070554937134618609926277437895029980245174
01474719468068984349536087237870814923058804265001775440222136692
19497268319149971553957338283899722324260346170316327132892172432
93414823219221781561202067498414863282586486396494894086735311984
87542808513750059732993808185407922249214024344950525276107816857
04707717662079906664246810240363777462148167179131661698526525933
27455038156208677911356439404008565505518270407112444336214476635
90179341953108431111005767617305509479336875319574363889314007557
80075653313610206913250864729950372237487765836958630210102788727
36316538995288936776915812144955490067485142591851239438281782980
49266403870939263057525714093639423600887115586024837895236645603
39309565092104096166270212323904043359754209734891624081548291170
80360665817900770688282037236785560524155682291636091743772117655
59023604955032646389401878907537621901104165756011023874877122701
96250964612099585830958920626319449976949804992786372992220222097
60635584717392117583345925369515540556572166535535975903825177497
033726681165539444448225642477607711558401523711

renders these 7,500 digits in binary:

This is a 30,000-digit prime:

And this is self-explanatory:

More here.

(Thanks, Gilles.)

In the 1960s, Soviet mathematician Vladimir Arnold mapped the square image of a cat to a torus, “stretched” (sheared) it as shown on that surface, then sliced the resulting image into pieces and recomposed them into a square.

As the process is repeated, any two points in the image quickly become separated, but, surprisingly, after sufficient repetitions the original image reappears.

A discrete analogue is at right. As the transformation is repeated, the image appears increasingly random or disordered, but the underlying cat can be glimpsed making occasional appearances, sometimes as a ghostly suggestion, sometimes in multiple smaller images, and occasionally (yowling, one imagines) even upside down.

It reappears again, unhurt, at the 300th iteration.

It’s called Arnold’s cat map. You can try it yourself here.

In his 1966 book New Mathematical Diversions From Scientific American, Martin Gardner predicted that the millionth digit of π would be 5. (At the time the value was known only to about 10,000 decimal places.) He was reprinting a column on π from 1960 and included this in the addendum:

It will probably not be long until pi is known to a million decimals. In anticipation of this, Dr. Matrix, the famous numerologist, has sent me a letter asking that I put on record his prediction that the millionth digit of pi will be found to be 5. His calculation is based on the third book of the King James Bible, chapter 14, verse 16 (it mentions the number 7, and the seventh word has five letters), combined with some obscure calculation involving Euler’s constant and the transcendental number e.

He’d intended this as a hoax, but eight years later the computers discovered he was right.