Pat'sBlog

Friday, 6 February 2026

Before there were Four-Fours, there were four threes, and several others

*Eyegate Gallery

EVERYONE has encountered the four-fours problem, using four fours and whatever mathematical operations that were allowed to make a number, or a set of numbers. You may even have read that it originated in the famous book of recreations by W. W. Rouse Ball; Wikipedia still has, "The first printed occurrence of this activity is in 'Mathematical Recreations and Essays' by W. W. Rouse Ball published in 1892. In this book it is described as a 'traditional recreation'. "

I know you've heard it before, but here we go again, "Wikipedia is wrong about that."

The first record I have found of a puzzle like these was in an 1818 edition of The schoolmaster's assistant: being a compendium of arithmetic both practical and theoretical : in five parts, and early American Arithmetic by Thomas Dilworth.

This image is from page 189 and part of a collection of "Short and Diverting Questions". As is typical of many of the early such problems, there were no specifications for the operations that might be employed. I have found the same exact problem in the 1800 edition.
In the same collection of problems, Dilworth poses a problem requesting the use of four threes...(which should give you a big clue if you are stuck on the previous problem of using four figures to make 12.

Ok, even I can do that one, and the dd+d/d format becomes a regular problem through the years with different digits; the most common being in the form of "use four nines to make 100."

Professor Singmaster says that both the Dilworth problems appear in a 1743 edition of this book.
By 1788 similar problems show up in another classic American Arithmetic by Nicolas Pike,"Said Harry to Edmund, I can place four 1's so that, when added, they shall make precisely 12. Can you do so too?"

The first printed version I can find of a question like this that asks about using three or four of the same number to find a set of integers appears in 1881 in a U.K. magazine called, Knowledge: an Illustrated Magazine of Science. It was founded and edited by Richard A Proctor, the English astronomer who is remembered for his maps of Mars (and has a crater there named for him). It may be that the Cupidus Scientiae who submitted the question is, in fact, the editor.

The next edition (Jan 6, 1882) did indeed carry the solutions, as well as correspondence from an H. Snell who provides that 19 = 4! - 4 - 4/4 (I have used the conventional current symbol for factorial but Snell used the Jarret symbol for factorial which looks like a right angle symbol with the number on the horizontal line and was popular at that time.)
The editor felt that factorials were inappropriate for the problem as posed. The following week (Jan 13,1882) there were several solutions for 19 from contributors, including (4+4-.4)/.4.

When W. W. Rouse Ball got into the act, it was in the third edition of his MRE 1896 and it was a long step away from the four-fours as we have come to know it. He repeated a problem previously used by Sam Loyd in 1893 which became popular in the United Kingdom; "Make 82 with the seven digits 9, 8, 7, 6, 5, 4, 0." Loyd offered a prize of 100 pounds for the solution. The solution, involving the use of repeating fractions, was given as 80.5 + .97 + .46 = 82 with all the decimal values repeating. This was indicated in the period by using a single dot above the values which repeated.

It was not until the fifth edition of 1911 of MRE that Ball gives the more common version, and describes it as, "An arithmetical amusement, said to have been first propounded in 1881,...) which seems to refer to the posting in Knowledge. By the sixth edition (1914) he has extended the problem to four nines and four threes. This one is significant because it seems to be the first that discusses what values can be achieved by what set of operations.

After a while it even caught on with higher level mathematicians. In 1991 Clifford A. Pickover asked for good approximations to Phi, (oft called the golden mean or golden ratio) using four fours.

$ \Phi = \frac{\sqrt(4) + \sqrt (4! - 4)}{4} = \frac{1 + \sqrt(5)}{2}$

A problem on Stack Exchange asked for four-fours to make pi. Maybe send your best response to these and your other favorite math constants. I'll keep updating with the good ones. (The really great ones I'll act like I thought of them first,) Use any operations that are commonly known.

Here is a short approach, π=arccos((4−4)− (4/4)) = Arccos (-1) .... because cos(π)= -1

In 1999 it became popular to ask for integers created using the digits 1, 9, 9, 9.
And it seems I saw a few of those floating around the internet at the beginning of 2012.

maybe you can rake a shot using 2,0,2,3 and see how many you can get with your choice of operations. I can get 21 and 25 quickly, the rest are up to you folks.

But remember, it all started with Jack and Harry, and four-threes.

On This Day in Math - February 6


Newton Statue - Trinity Chapel, Cambridge UK

The feeling of it (pure oxygen) to my lungs was not sensibly different from that of common air; but I fancied that my breast felt peculiarly light and easy for some time afterwards. Who can tell but that, in time, this pure air may become a fashionable article in luxury. Hitherto only two mice and myself have had the privilege of breathing it.~Joseph Priestley

The 37th day of the year; 37 is the only prime with a three digit period for the decimal expansion of its reciprocal, 1/37 = .027027.... But 37 has a strange affinity with 27, which also has a three digit period for its reciprocal, .037037..., The affinity, of course, is due to 27 x 37 = 999

Big Prime::: n = integer whose digits are (left to right) 6424 copies of 37, followed by units digit of 3, is prime (n = 3737...373 has 12849 digits) *Republic of Math ‏@republicofmath

An amazing reversal: 37 is the 12th prime & 73 is the 21st prime . This enigma is the only known combination.

If you use multiplication and division operations to combine Fibonacci numbers, (for example, 4 = 2^2, 6 = 2·3, 7 = 21/ 3 ,...) you can make almost any other number. Almost, but you can't make 37. In fact, there are 12 numbers less than 100 that can not be expressed as "Fibonacci Integers" *Carl Pomerance, et al.

Any number divisible by 37 will still be divisible by 37 if you rotate its digits, 148, 481, 1nd 814 are all divisible by 37.

EVENTS

1672 Newton wrote Henry Oldenburg about his optical theories, (including the phrase, "because that Light is a heterogenous mixture of differently refrangible rays." and Oldenburg published them a few days later in the

Philosophical Transactions. The controversy that followed dissuaded Newton from publishing on optics—and also on the calculus—until 1704 *ISIS, 69, p 134 (*VFR)

But it is requisite, that the prism and lens be placed steady, and that the paper, on which the colours are cast be moved to and fro; for, by such motion, you will not only find, at what distance the whiteness is most perfect but also see, how the colours gradually convene, and vanish into whiteness, and afterwards having crossed one another in that place where they compound whiteness, are again dissipated and severed, and in an inverted order retain the same colours, which they had before they entered the composition. You may also see, that, if any of the colours at the lens be intercepted, the whiteness will be changed into the other colours. And therefore, that the composition of whiteness be perfect, care must be taken, that none of the colours fall besides the lens.

Some of his opponents denied the truth of his experiments, refusing to believe in the existence of the spectrum. Others criticized the experiments, saying that the length of the spectrum was never more than three and a half times the breadth, whereas Newton found it to be five times the breadth. It appears that Newton made the mistake of supposing that all prisms would give a spectrum of exactly the same length;

1766 Just a few months before he returns to St. Petersburg, Euler reads his paper (E401) “A New Method for Comparing the Observation of the Moon to Theory” to the Berlin Academy. The paper proposes numerical techniques for approximating a body's velocity and acceleration. Sandifer suggests that the paper had great influence on LaGrange’s foundational program for the Calculus. *Ed Sandifer, How Euler Did It, MAA

1828 George Biddell Airy appointed Plumian professor of astronomy at Cambridge at a salary of £500 per annum. He was appointed even after he raised a row that the previous salary of £300 was inadequate. For the previous two years he held the Lucasian professorship—the post Newton held—at a salary of £99. *VFR

1930 Kurt G¨odel received his Ph.D. from the University of Vienna for a dissertation, directed by Hans Hahn, that showed the completeness of ﬁrst order logic (every valid ﬁrst-order formula is provable). *VFR

Gödel's discoveries in the foundations of mathematics led to the proof of his completeness theorem in 1929 as part of his dissertation to earn a doctorate at the University of Vienna, and the publication of Gödel's incompleteness theorems two years later, in 1931. The incompleteness theorems address limitations of formal axiomatic systems. In particular, they imply that a formal axiomatic system satisfying certain technical conditions cannot decide the truth value of all statements about the natural numbers, and cannot prove that it is itself consistent.[4][5] To prove this, Gödel developed a technique now known as Gödel numbering, which codes formal expressions as natural numbers.

Gödel also showed that neither the axiom of choice nor the continuum hypothesis can be disproved from the accepted Zermelo–Fraenkel set theory, assuming that its axioms are consistent. The former result opened the door for mathematicians to assume the axiom of choice in their proofs. He also made important contributions to proof theory by clarifying the connections between classical logic, intuitionistic logic, and modal logic.

1935, the board game Monopoly went on sale under its present day name for the first time. Prior to being purchased by Parker Brothers, the game had been developed by Elizabeth Magie under the name, “The Landlord’s Game.” Magie intended the game to give an economic lesson about land value tax. Parker Brothers initially rejected the game, claiming that it was too complicated and took too long to finish, but later changed their opinion based on the game’s popularity in Pennsylvania.

Monopoly has grown into one of Parker Brothers’ most successful board game franchises.

*Famous Daily

1959 Kilby Files Patent For Integrated Circuit.
Jack Kilby of Texas Instruments files a patent application called "miniaturized electronic circuits" for his work on a multi-transistor device. The patent was only one of 60 that Kilby holds. While Kilby has the earliest patent on the "integrated circuit," it was Robert Noyce, later co-founder of Intel, whose parallel work resulted in a practical device. Kilby's device had several transistors connected by flying wires while Noyce devised the idea of interconnection via a layer of metal conductors. Noyce also adapted Jean Hoerni's planar technique for making transistors to the manufacture of more complex circuits. *CHM

Two drawings from Kilby's first IC patent *haverford.edu

BIRTHS

1465 Scipione del Ferro (6 February 1465 – 5 November 1526) born in Bologna, Italy. Around 1515 he solved the cubic equations x³+px = q and x³= px + q when p and q are positive. His methods are unknown. This information was passed on to his son-in-law Annibale dalla Nave who was tricked into revealing it to Cardano, who published it in his Ars magna of 1545.*VFR
There are no surviving scripts from del Ferro. This is in large part due to his resistance to communicating his works. Instead of publishing his ideas, he would only show them to a small, select group of friends and students. It is suspected that this is due to the practice of mathematicians at the time of publicly challenging one another. When a mathematician accepted another's challenge, each mathematician needed to solve the other's problems. The loser in a challenge often lost funding or his university position. Del Ferro was fearful of being challenged and likely kept his greatest work secret so that he could use it to defend himself in the event of a challenge.
Despite this secrecy, he had a notebook where he recorded all his important discoveries. After his death in 1526, this notebook was inherited by his son-in-law Hannival Nave, who was married to del Ferro's daughter, Filippa. Nave was also a mathematician and a former student of del Ferro's, and he replaced del Ferro at the University of Bologna after his death. In 1543, Gerolamo Cardano and Lodovico Ferrari (one of Cardano's students) travelled to Bologna to meet Nave and learn about his late father-in-law's notebook, where the solution to the depressed cubic equation appeared.
Del Ferro also made other important contributions to the rationalization of fractions with denominators containing sums of cube roots.
He also investigated geometry problems with a compass set at a fixed angle, but little is known about his work in this area. *Wik (Teachers may need to explain to students how suppression of the squared term allows this to solve general cubics.)

Four years before del Ferro died, the brilliant mind that would solve the quartic (fourth power) equation was born. His name was Lodovico Ferrari.

1695 Nicolaus II Bernoulli (February 6, 1695, Basel, Switzerland – July 31, 1726, St. Petersburg, Russia) was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. Nicolaus worked mostly on curves, differential equations, and probability. He was a contemporary of Leonhard Euler. He also contributed to fluid dynamics.*Wik He was the oldest and favorite of three sons of Johann Bernoulli. He made important mathematical contributions to the problem of trajectories while working on the mathematical arguments behind the dispute between Newton and Leibniz.*SAU When the father was asked to come to St. Petersburg to join the Academy, he declined because of his age. He suggested that they take his son Nikolaus, but, so that he not be lonely, they should also take another son Daniel. Unfortunately, Nikolaus II drowned in 1726, only eight months after going to St. Petersburg. His professorship was succeeded in 1727 by Leonhard Euler, whom the Bernoulli brothers had recommended.

1802 Sir Charles Wheatstone, (6 Feb 1802, 19 Oct 1875) English physicist who popularized the Wheatstone bridge, a device that accurately measured electrical resistance and became widely used in laboratories. He didn't actually invent the "Wheatstone Bridge". His contemporary, Samuel Hunter Christie, came up with the idea of the bridge circuit, but Wheatstone set the precedent for using it in the way in which it has been most commonly used. Over time, the device became associated with him and took on his name. He did, however, invent the concertina (1829), the stereoscope (1838), and an early form of the telegraph. He also developed a chronoscope (1842) to determine the velocity of projectiles at an English gunnery.*TIS (For students of discrete math, or interested in codes, Wheatstone was also the creator of the Playfair Cipher.) {Wheatstone's work was so diverse that after a lecture at the Science Conference in South Kensington (London) by Prof. W. G. Adams on Wheatstone's acoustical discoveries, William Spottiswoode commented, "It must have struck all those in science... that when they fancied they had found something new, they find it was done by Sir Charles Wheatstone years ago." *Knowledge and Scientific News, Jan 1908, pg 7

1824 William Huggins (7 February 1824 – 12 May 1910) was an English amateur astronomer. He built his own private observatory, called Tulse Hill, in London in 1856, and when spectroscopy was established as a scientific field of inquiry in 1859, Huggins jumped at the chance to apply it to astronomy. He said he felt like a parched man spying water in the desert. In the early 1860s, he examined the light of several nebulae and found that the spectral lines were those of a gas, not stars , and suggested that at least some nebulae were gaseous in nature (at the time, no one knew what nebulae were, except that they were nebulous).

In 1868, Huggins discovered that the spectral lines of the star Sirius are shifted slightly to the red end of the spectrum, which he recognized as a indication that Sirius is moving away from us, and he even measured its speed of recession. Huggins's wife was his close collaborator throughout his career, and just before he died, he and Lady Huggins republished all his scientific papers in a beautiful volume, The Scientific Papers of Sir William Huggins (1909) *Linda Hall Org

1875 Joseph Winlock (February 6, 1826 – June 11, 1875) was an American astronomer and mathematician.

He was born in Shelby County, Kentucky, the grandson of General Joseph Winlock (1758–1831). After graduating from Shelby College in Kentucky in 1845, he was appointed professor of mathematics and astronomy at that institution.

From 1852 until 1857 he worked as a computer for the American Ephemeris and Nautical Almanac, and relocated to Cambridge, Massachusetts. He briefly served as head of the department of mathematics at the United States Naval Academy, but returned as superintendent of the Almanac office. He was elected a Fellow of the American Academy of Arts and Sciences in 1853.

He married Isabella Washington in Shelbyville, Kentucky on December 10, 1856, and they had six children.

In 1863 he was one of the fifty charter members of the National Academy of Sciences.[4] Three year later in 1866 he became director of the Harvard College Observatory, succeeding George Bond, and making many improvements in the facility. He was also appointed professor of astronomy at Harvard. He remained at the university, eventually becoming professor of geodesy until his sudden death in Cambridge on June 11, 1875.

Much of his astronomical work included measurements with the meridian circle, a catalogue of double stars and stellar photometry investigations. He also led solar eclipse expeditions to Kentucky in 1860 and Spain in 1870.

The crater Winlock on the Moon is named after him.

1848 Adam Wilhelm Siegmund Günther (6 Feb 1848 in Nuremberg, Germany - 3 Feb 1923 in Munich, Germany) Günther's contributions to mathematics include a treatise on the theory of determinants (1875), hyperbolic functions (1881), and the parabolic logarithm and parabolic trigonometry (1882). He also wrote numerous books and journal articles [which] encompass both pure mathematics and its history and physics physics, geophysics, meteorology, geography, and astronomy. The individual works on the history of science, worth reading even today, bear witness to a thorough study, a remarkable knowledge of the relevant secondary literature, and a superior descriptive ability. *SAU

1900 Rosalind Cecilia Hildegard Tanner (née Young) (5 February 1900 – 24 November 1992) was a mathematician and historian of mathematics. She was the eldest daughter of the mathematicians Grace and William Young. She was born and lived in Göttingen in Germany (where her parents worked at the university) until 1908. During her life she used the name Cecily.

Cecily joined the University of Lausanne in 1917. She also helped her father's research between 1919 and 1921 at the University College Wales in Aberystwyth, and worked with Edward Collingwood, also of Aberystwyth, on a translation of Georges Valiron's course on Integral Functions. She received a L-És-sc (a bachelor's degree) from Lausanne in 1925.

She then studied at Girton College, Cambridge, gaining a PhD in 1929 under the supervision of Professor E. W. Hobson for research on Stieltjes integration. She accepted a teaching post at Imperial College, London where she worked until 1967.

After 1936, most of her research was in the history of mathematics, and she had a particular interest in Thomas Harriot, an Elizabethan mathematician. She set up the Harriot Seminars in Oxford and Durham. Rosalind married William Tanner in 1953; however, he died a few months after their marriage.

In 1972 she and Ivor Grattan-Guinness published a second edition of her parents' book The Theory of Sets of Points, originally published in 1906.

Mrs. R. C. H. Young (left, upper) at the ICM 1932

1916 John Crank (6 February 1916 – 3 October 2006) was a mathematical physicist, best known for his work on the numerical solution of partial differential equations.
He worked on ballistics during the Second World War, and was then a mathematical physicist at Courtaulds Fundamental Research Laboratory from 1945 to 1957. In 1957, he was appointed as the first Head of Department of Mathematics at Brunel College in Acton. He served two terms of office as Vice-Principal of Brunel before his retirement in 1981, when he was granted the title of Professor Emeritus.
Crank's main work was on the numerical solution of partial differential equations and, in particular, the solution of heat-conduction problems. He is best known for his work with Phyllis Nicolson on the heat equation, which resulted in the Crank–Nicolson method.*Wik

DEATHS

1612 Christopher Clavius (March 25, 1538 – February 6, 1612 {some sources give Feb 12 for the date of death}), the Euclid of the sixteenth-century, born in the German town of Bamberg, the see of the prince-bishop of Franconia. He was also the leader of the Gregorian calendar reform. Perhaps his greatest contribution was as an educational reformer. *Renaissance Mathematicus He was a German Jesuit mathematician and astronomer who was the main architect of the modern Gregorian calendar. In his last years he was probably the most respected astronomer in Europe and his textbooks were used for astronomical education for over fifty years in Europe and even in more remote lands (on account of being used by missionaries). As an astronomer Clavius held strictly to the geocentric model of the solar system, in which all the heavens rotate about the Earth. Though he opposed the heliocentric model of Copernicus, he recognized problems with the orthodox model. He was treated with great respect by Galileo, who visited him in 1611 and discussed the new observations being made with the telescope; Clavius had by that time accepted the new discoveries as genuine, though he retained doubts about the reality of the mountains on the Moon. Later, a large crater on the Moon was named in his honour. *Wik

Stephen Gray (December 1666 – 7 February 1736) was an English dyer and astronomer who was the first to systematically experiment with electrical conduction. Until his work in 1729 the emphasis had been on the simple generation of static charges and investigations of the static phenomena (electric shocks, plasma glows, etc.). Gray showed that electricity can be conducted through metals and that it appeared on the surfaces of insulators.

Gray went on to make more electrical discoveries, the most noticeable being electrical induction (creating an electrical charge in a suspended object without contact). In 1729 Gray showed that the "Electrik Vertue" of a glass tube can be conducted via a metal to another body. This was the first evidence that electricity was not a property of the glass tube but some kind of fluid. That same year Gray create two oaken cubes, one solid and one hollow; observing that these had the same electrical properties demonstrated that electricity was a surface rather than a bulk character.

This experiment was widely celebrated around Europe as the famous "Flying Boy" demonstration: a boy was suspended on silk cords, and then charged by Gray bringing his rubbed tube (static electric generator) close to the boy's feet, but without touching. Gray showed that the boy's face and hands still attracted the chaff, paper and other materials. Gray noted the crackling of 'electric virtue' resembled lightning (as did other experimenters), foreshadowing the great discoveries of Benjamin Franklin nearly a century later.

Stephan Gray's “electric boy” experiment, in which a boy hanging from insulating silk ropes is given an electric charge. A group are gathered around. A woman is encouraged to bend forward and poke the boy's nose, to get an electric shock.*Wik

1804 Joseph Priestley (13 Mar 1733, 6 Feb 1804) English chemist, clergyman and political theorist who discovered the element oxygen. His early scientific interest was electricity, but he is remembered for his later work in chemistry, especially gases. He investigated the "fixed air" (carbon dioxide) found in a layer above the liquid in beer brewery fermentation vats. Although known by different names at the time, he also discovered sulphur dioxide, ammonia, nitrogen oxides, carbon monoxide and silicon fluoride. Priestley is remembered for his invention of a way of making soda-water (1772), the pneumatic trough, and recognizing that green plants in light released oxygen. His political opinions and support of the French Revolution, were unpopular. After his home and laboratory were set afire (1791), he sailed for America, arriving at New York on 4 Jun 1794 *TIS He died on the morning of 6 February 1804 and was buried at Riverview Cemetery in Northumberland, Pennsylvania.

Priestley's epitaph reads:
Return unto thy rest, O my soul, for the
Lord hath dealt bountifully with thee.
I will lay me down in peace and sleep till
I awake in the morning of the resurrection. *Wik

*cometography.com

1923 Edward Emerson Barnard (16 Dec 1857; 6 Feb 1923) astronomer who pioneered in celestial photography, specializing in wide-field photography. From the time he began observing in 1881, his skill and keen eyesight combined to make him one of the greatest observers. Barnard came to prominence as an astronomer through the discovery of numerous comets. In the 1880s, a patron of astronomy in Rochester, N.Y. awarded $200 per new comet was found. Barnard discovered eight - enough to build a "comet house" for his bride. At Lick Observatory (1888-95) he made the first photographic discovery of a comet; photographed the Milky Way; and discovered the fifth moon of Jupiter. Then he joined Yerkes Observatory, making his Photographic Atlas of Selected Regions of the Milky Way.*TIS The faint Barnard's Star is named for Edward Barnard after he discovered in 1916 that it had a very large proper motion, relative to other stars. This is the second nearest star system to the Sun, second only to the Alpha Centauri system. *Wik

1965 Ernst Erich Jacobsthal (16 October 1882, Berlin – 6 February 1965, Überlingen) was a German mathematician, and brother to the archaeologist Paul Jacobsthal.
In 1906, he earned his PhD at the University of Berlin, where he was a student of Georg Frobenius, Hermann Schwarz and Issai Schur; his dissertation, Anwendung einer Formel aus der Theorie der quadratischen Reste (Application of a Formula from the Theory of Quadratic Remainders), provided a proof that prime numbers of the form 4n + 1 are the sum of two square numbers. *Wik The theory was first conjectured by Fermat and proved by Euler.

1973 Ira Sprague Bowen (21 Dec 1898; 6 Feb 1973) was an American astrophysicist. His investigation of the ultraviolet spectra of highly ionized atoms led to his explanation of the unidentified strong green spectral lines of gaseous nebulae (clouds of rarefied gas) as forbidden lines of ionized oxygen and nitrogen. This emission, appearing to match no known element, had formerly been suggested to be due to a hypothetical element, "nebulium." Bowen was able to show, that in reality, the emission lines exactly matched those calculated to be the "forbidden lines" of ionized oxygen and nitrogen under extremely low pressure. This made a major advance in the knowledge of celestial composition. He was director of the Mt. Wilson and Palomar Observatories from 1948-64.*TIS

Ira Bowen (right) in the lab with Robert Millikan.

1992 Caius Jacob (29 March 1912 , Arad - 6 February 1992 , Bucharest ) was a Romanian mathematician and member of the Romanian Academy. He made contributions in the fields of fluid mechanics and mathematical analysis , in particular vigilance in plane movements of incompressible fluids, speeds of movement at subsonic and supersonic , approximate solutions in gas dynamics and the old problem of potential theory. His most important publishing was Mathematical introduction to the mechanics of fluids. *Wik

2017 Raymond Merrill Smullyan ( May 25, 1919 -February 6, 2017) is an American mathematician, concert pianist, logician, Taoist philosopher, and magician. His first career (like Persi Diaconis a generation later) was stage magic. He then earned a BSc from the University of Chicago in 1955 and his Ph.D. from Princeton University in 1959. He is one of many logicians to have studied under Alonzo Church. Smullyan is the author of many books on recreational mathematics, recreational logic, etc. Most notably, one is titled "What Is the Name of This Book?". *Wik For example the book is described on the cover as follows:"Beginning with fun-filled monkey tricks and classic brain-teasers with devilish new twists, Professor Smullyan spins a logical labyrinth of even more complex and challenging problems as he delves into some of the deepest paradoxes of logic and set theory, including Gödel's revolutionary theorem of undecidability."
Martin Gardner described this book in Scientific American as:"The most original, most profound and most humorous collection of recreational logic and mathematics problems ever written."

Credits :
*CHM=Computer History Museum
*FFF=Kane, Famous First Facts
*NSEC= NASA Solar Eclipse Calendar
*RMAT= The Renaissance Mathematicus, Thony Christie
*SAU=St Andrews Univ. Math History
*TIA = Today in Astronomy
*TIS= Today in Science History
*VFR = V Frederick Rickey, USMA
*Wik = Wikipedia
*WM = Women of Mathematics, Grinstein & Campbell

Thursday, 5 February 2026

Matrices and Magic Squares

Matrices and Magic Squares

In the last half of January a few years back, John Cook posted a blog about Matrices made up of Magic Squares. He pointed out that if you multiply an odd number of 3 × 3 magic squares together, the result is a magic square. He used the three Spanish Magic Squares above from another of his posts as an example. The conjecture is that it would work for squares of any order, but that may not have been proven yet.

Since -1 is an odd number, it followed that the inverse of a magic square matrix would form a magic square also, so I gave it a go on Wolfram Alpha. The oldest and most common magic square known is the one with integers from 1 to 15 with a total for each row, column, and diagonal of 15 (and five in the center square).

And the inverse came out to be:

If you add up any row, column, or diagonal you should get $ \frac{24}{360} = \frac{1}{15}$ which we might expect, since the product of the inverses has a determinant of one. To make that happen the sequence of numbers 1 to 9 in the original became the sequence from $ \frac{-32}{360}$ and increasing by $ \frac {1}{15} $ each step until it reaches the highest value. The center number, as in any 3x3 magic square must be 1/3 of the total.

If you add up all the numbers in the original magic square, you get 45. If you add up all the numbers in the inverse, you get 1/5, or nine times the center value of 8/360?

If this was consistent in all such inverses of magic square matrices, ... we might expect that in a 5x5 matrix inverse would produce a sum of all the entries that is 1/13? As it turned out, it did. And the sum of each row, column, etc would be 1/65. The center term of the 5x5 should then be 1/5 of 1/65, or 1/325, with the 12 numbers above it increasing by 1/13 each, and the 12 below it decreasing 1/13 each. But when I did the inverse of a 5x5 on Wolfram Alpha, the values were not in the same order as the originals. For example, the smallest number in the standard 3x3, 1, is in the same position as the smallest number in the inverse, -52/360; but when I did the 5x5, the smallest number was NOT in the same spot as the one, and more importantly, their were numbers that repeated.

Even though the method of placing 1/325 in the center square with increments in order as indicated above will produce a magic square with all the same values suggested for rows, columns, totals, etc, it simply is not the inverse of the classic 5x5 magic square. The idea that the inverse would produce a magic square is correct, but not in the classic style in which each value is unique and the numbers all form a sequence. I have not gone on to see if other NxN magic square matrices might have a similar affliction, but would enjoy you sharing your results.

A similar conflict may happen when you cube a magic square matrix. While the 3x3 comes out as expected, the 5x5 has duplicate values in the upper left and lower right corners.

On This Day in Math - February 5

See Events:1897

It is a mathematical fact that the casting of this pebble from my hand alters the center of gravity of the universe.
~Thomas Carlyle

In 2017 on this date was the 5th day of the 2nd month, in 2017 teachers can offer their students, $Sin(2017 \sqrt[5]{2}) = -1 $ (It's not, but it will give that answer on the Ti-84 calculators (mine does at least.) The true answer is

The 36th day of the year; 36 is the smallest non trivial number which is both triangular and square. It's also the largest day number of the year which is both. What's the next? You can find an infinity of them using this beautiful formula from Euler, Hat Tip to Vincent PANTALONI @panlepan

36 is the sum of the first three cubes, $1 ^3 + 2^3 + 3^3 = 36$ The sums of the first n cubes is always a square number. $\sum_{k=1}^n k^3 = (\frac{(n)(n+1)}{2}) ^2$ Note that this sequence and its formula were known to (and possibly discovered by) Nicomachus, 100 CE) (There are only two year days that are square numbers that are the sum of three distinct cubes, can you find the square, and ir's cube partitions)

The sum of the first 36 integers, $\sum_{k=1}^{36} k = 666$ the so called "number of the beast."

And Mario Livio pointed out in a tweet that this is 5/2 in European style dating, and 52 is the maximum number of moves needed to solve the "15" sliding puzzle from any solvable position.

The length of an Inch was decreed by King Edward in 1324 to be “three grains of barley, dry and round, placed end to end lengthwise." Which makes a foot the length of 36 such barleycorns.

The Kiwi's seeds divide the circle into 36 equal sections. Nature's protractor. *Matemolivares@Matemolivares

A special historical tribute to 36: The thirty-six officers problem is a mathematical puzzle proposed by Leonhard Euler in 1782. He asked if it were possible to place officers of six ranks from each of six regiments in a 6x6 square so that no row or column would have an officer of the same rank, or the same regiment. Euler suspected that it could not be done. Euler knew how to construct such squares for nxn when n was odd, or a multiple of four, and he believed that all such squares with n = 4m+2 (6, 10, 14...) were impossible ( Euler didn't say it couldn't be done. He just said that his method does not work for numbers of that form.) Proof that he was right for n=6 took a while. French mathematician (and obviously a very patient man) Gaston Tarry proved it in 1901 by the method of exhaustion. He wrote out each of the 9408 6x6 squares and found that none of them worked. Then in 1959, R.C. Bose and S. S. Shrikhande proved that all the others could be constructed. So the thirty-six square is the only one that can't be done.

Orthogonal Latin squares have been known to predate Euler. As described by Donald Knuth the construction of 4x4 set was published by Jacques Ozanam in 1725 (in Recreation mathematiques et physiques, Vol. IV) as a puzzle involving playing cards. The problem was to take all aces, kings, queens and jacks from a standard deck of cards, and arrange them in a 4x4 grid such that each row and each column contained all four suits as well as one of each face value. This problem has several solutions known back to 1725, but no one could figure out how many in all.

A common variant of this problem was to arrange the 16 cards so that, in addition to the row and column constraints, each diagonal contains all four face values and all four suits as well.

According to Martin Gardner, who featured this variant of the problem in his November 1959 Mathematical Games column, the number of distinct solutions was incorrectly stated to be 72 by Rouse Ball. This mistake persisted for many years until the correct value of 144 was found by Kathleen Ollerenshaw. Notice both 72 and 144 are multiples of 36.

EVENTS

1575 Jan De Groot entered the University of Leiden, in the Netherlands, on its opening day. With Simon Stevin he later performed an experiment proving that bodies of different weights fall the same distance in the same time (published 1586 by Stevin). This anti-Aristotelian experiment anticipated Galileo’s famous, but apocryphal, experiment at the Leaning Tower of Pisa. His son Hugo De Groot was a famous jurist. *VFR Thony Christie pointed out that "The anti-Aristotle tower and ball experiment was first carried out by Johannes Philiponus in 6th century CE". Philiponus proposed a kinetic theory for motion in place of Aristotle's impetus.

1673 Robert Hooke writes in his journal that he had, "Told the Society of Arithmetick engine.‏*@HookesLondon It is said that Newton had this, and other Hooke items, including Hooke's portrait, removed from the Royal Society after Hooke's death but this does not seem to be supported by most math historians

5 Feb 1675 (OS) 15 Feb 1676(NS) Newton wrote Hooke: "What DesCartes did was a good step....If I have seen further it is by standing on ye sholders of Giants." *VFR
The letter is at the Historical Society of Pennsylvania.

1689 The Convention Parliament, with Cambridge U. MP Isaac Newton voting in the majority, declared the throne of England vacant after James II escaped to France with the permission of his Son-in-Law and daughter, William and Mary, who were offered the crown jointly. The only record of a comment by Newton during the Parliament except to ask for a servant to close a drafty window. *Thomas Levenson, Newton and The Counterfeiter.

1772 Laplace presented his ﬁrst probability memoir,, titled "Mémoire sur la probabilité des causes par les événements" (Memoir on the Probability of Causes from Events), to the Académie des Sciences, VFR (sometimes cited in 1774?)

This paper is considered a foundational work in the field of probability theory, as it marked Laplace's early contributions to the subject and introduced important concepts like inverse probability.

A PDF file of S M Stigler's translation is here.

1796 Schiller (1759–1815) wrote to Goethe (1749–1832): “Wo es die Sache leidet, halte ich es immer f¨ur besser, nicht mit dem Anfang anzufangen, der immer das Schwerste ist.” (I always think it better, whenever possible, not to begin at the beginning, as it is always the most difficult part). Although this is advice from one poet to another, it seems to apply to mathematics, especially the foundations of mathematics. Quoted from Numbers (1990) by H.-D. Ebinghaus et al., p. 6. *VFR

1835 A ceremony to honor "The Genius and Discoveries of Sir Isaac Newton" was organized by the citizens of the Lincolnshire, his area of birth, a few years after the centennial of his death. By unanimous choice, the committee selected as the speaker, the 19 yr old George Boole. "All present were struck by the youthful age of the speaker and not a little amazed by both his knowledge of the subject and his confident lecturing style."
*Desmond MacHale, The Life and Work of George Boole

1840 The American Statistical Association held its ﬁrst annual meeting, in Boston. "On November 27, 1839, five men held a meeting in the rooms of the American Education Society at No. 15 Cornhill in Boston, Massachusetts, to organize a statistical society. Its purpose, as stated in the society's first constitution, was to "collect, preserve, and diffuse statistical information in the different departments of human knowledge." Originally called the American Statistical Society, the organization's name was changed to the American Statistical Association (ASA) at its first annual meeting, held in Boston on February 5, 1840. " *Robert L. Mason, ASA: The First 160 Years

The phrase “timing is everything” rings true for the American Statistical Association (ASA), which was formed in November 1839 in Boston—a city that was burgeoning with educational and technical professional opportunities. Back then, annual dues cost $2, and notable figures who joined the society included US President Martin Van Buren and Minister to France Lewis Cass. In a then male-dominated environment, Florence Nightingale—well known for her contributions to the medical field—also became a member and was recognized for using statistical analysis techniques in her data-collection efforts benefitting public health and welfare.

What started with just five founders soon began to attract hundreds of members who were elected, not simply granted voluntary membership. With ties to statistical work of the US government, particularly the Census Bureau, a movement was led by then-ASA President Francis A. Walker (1883–1897) to expand the association’s roots nationwide. ASA leadership chose to drop the exclusive “by invitation only” membership requirement to attract a larger, more diverse group of members and opened the doors for professionals from a variety of fields and business sectors.

The organization began publishing the Journal of the American Statistical Association (JASA) in 1888, when its mission was solidified and membership was growing. And in 1908, the ASA conducted its annual meeting in Atlantic City, New Jersey—stepping away from the traditional stronghold of Boston.

1843 The great comet of 1843, A night-time view showing an eyewitness account of the Great Comet of 1843, painted by the astronomer Charles Piazzi Smyth. The earliest observation occurred on the evening of 5 of February, 1843 and Smyth recorded its appearance at the Royal Observatory, Cape of Good Hope, South Africa between 3 and 6 of March. When at its greatest brilliance, it was visible only from southern latitudes. The view in the painting is probably taken from the Observatory. It shows Table Bay with Table Mountain visible in the background on the left. A large sailing ship sits in the foreground on the right, with other shipping in the distance. One of the great British astronomers, Smyth was 42 years Astronomer Royal for Scotland. *Royal Museums Greenwich

1850 D. D. Parmalee issued a patent (US Patent # 7074) for the ﬁrst key-driven adding machine. *VFR

While this was the first US patent, an earlier key-driven machine had been patented "as early as 1844 by Jean-Baptiste Schwilgue´ (1776– 1856), together with his son Charles. Jean-Baptiste Schwilgue´ was the architect of Strasbourg’s third astronomical clock during the years 1838–1843. He was trained as a clockmaker, but also became professor of mathematics,weights and measures controller, and an industry man, whose particular focus was on improving scales." *Denis Roegel, An Early (1844) Key-Driven Adding Machine, IEEE Annals of the History of Computing, Volume 30, Number 1, January-March 2008, pp. 59-65

In 1897, the Indiana State House legislature presented Bill No.246 which in effect gave 3.2 exactly as the value of pi. It stated, in part, "the ratio of the diameter and circumference [pi] is as five-fourths to four." That is (4 divided by 5/4) = 16/5 = 3.2 exactly. It was introduced by Representative Taylor I. Record, a farmer and lumber merchant, on behalf of a mathematical hobbyist, Dr. Edwin J. Goodwin, M.D. Neither they, nor the House politicians, understood it was mathematically incorrect. That was shortly recognized by Clarence A. Waldo, mathematics professor at Purdue University, who advised the Indiana Senators. They indefinitely postponed the bill on 12 Feb 1897. Pi is, in fact, an irrational number, approx. 3.141592.*TIS

The Committee on Education, reported favorably and following a motion to suspend the rules, the bill passed on February 6, 1897 without a dissenting vote. Although called the "Pi" bill, it did not use the name. The bill also reported that the square root of two would be 10/7. *Wik

1901 Loop-the-loop centrifugal RR (roller coaster) patented by Ed Prescot. (I have also seen the date of patent as August 16, 1898. This date is now the National Roller Coaster Day in the US. ) Prescott,an inventor and mechanic from Arlington, Massachusetts. Prescott’s Loop the Loop coaster, a dual-tracked steel roller coaster, was installed at Coney Island, New York from 1901 to 1910. It seems more people came to look, than to ride.

No more looping roller coasters were built until 1976 when Revolution opened at Six Flags Magic Mountain.*Wik

The vertical loop is not a recent roller coaster innovation. Its origins can be traced back to the 1850s when centrifugal railways were built in France and Great Britain. In 1901 Prescott built the Loop-the-Loop at Coney Island.

*Smithsonian Mag

1920 Discussion on the theory of relativity by J. H. Jeans – a meeting of the Royal Society, *Royal Society Journal, HT Katharina H Mathsbooks

1924 The Royal Greenwich Observatory begins broadcasting the time "pips" on BBC, a series of six short tones broadcast at one-second intervals by many BBC Radio stations. The pips were introduced in 1924 and have been generated by the BBC since 1990. The pips were the idea of the Astronomer Royal, Sir Frank Watson Dyson, and the head of the BBC, John Reith.*Wik
This eight-day wall-mounted astronomical regulator by Edward John Dent & Co was originally made for use in observing the Transit of Venus in 1874. In 1923 it was adapted as the primary standard for the new six-pip time signal. The clock sent electrical impulses down a telephone wire to the BBC for conversion into audio pips for radio broadcasts. It has a zinc tube temperature-compensated pendulum and was corrected from 1929 by the Shortt master clock number 16. The three sets of contacts for closing the six-pip circuit every quarter of an hour can be seen in two of the holes within the seconds dial, and halfway down the pendulum, operated by a roller. This clock was in service for the BBC signal at the Observatory from 1924 to 1949, when it was superseded by a quartz clock. *Royal Observatory Greenwich

1930 You hear all those "Einstein said, " quotes and you wonder which ones are true, knowing most of them are true. And the one about, "Life is like riding a bicycle...", well it is mostly true, but the almost always muck it up a little, like the one in the image below. Einstein wrote it in a letter to his son, Eduard, on February 5, 1930, trying to encourage his son, he wrote "Men are like bicycles, It's only easy to keep your balance when you are on the move".

This letter and one more written to his son on January 24 of the same year, both signed "Papa" , were sold at Christie"s on June 7, 2000 for 4935 GBP.

1958 Kilby Files a Patent for the Integrated Circuit. Jack Kilby of Texas Instruments files a patent application called miniaturized electronic circuits for his work on a multi-transistor device. The patent was only one of 60 that Kilby holds. While Kilby has the earliest patent on the integrated circuit, it was Robert Noyce, later co-founder of Intel, whose parallel work resulted in a practical device. Kilby's device had several transistors connected by flying wires while Noyce devised the idea of interconnection via a layer of metal conductors. Noyce also adapted Jean Hoerni's planar technique for making transistors to the manufacture of more complex circuits. *CHM

*Wik

In 1962, the Sun, the Moon, and the five naked-eye visible planets - Mercury, Venus, Mars, Jupiter, and Saturn - were in conjunction. Though not in a straight line along their orbital paths, as viewed in the sky, they were within 16 degrees of each other (meaning all appeared within a circle just 16 º across). This conjunction coincided with a total solar eclipse, which made viewing Mercury, Venus, Mars, Jupiter, and Saturn possible for a brief period of time from a small stretch of Earth where the eclipse's shadow hit. The five naked-eye visible planets cluster together in the sky within a circle 25 degrees or less in diameter once every 57 years, on average. The next time in the 21st century that this will happen is 8 Sep 2040. *TIS (image...In May of 2011 a planetary conjunction of Mercury, Venus, Mars and Jupiter appeared very close to each other in the sky.) And for St. Valentines day this year (2012) I have ordered up a conjunction with Mercury and Neptune less than 1.5 ^o apart for my beautiful Jeannie, but the rest of you may enjoy it as well.

1974 US Mariner 10 returns 1st close-up photos of Venus' cloud structure2040 The near-Earth asteroid 2011 AG5 currently has an impact probability of 1 in 625 for Feb. 5, 2040, according to Donald Yeomans, head of the Near-Earth Object Observations Program at NASA’s Jet Propulsion Laboratory in Pasadena, California. Made using an ultraviolet filter in its imaging system, the photo has been color-enhanced to bring out Venus's cloudy atmosphere as the human eye would see it. Venus is perpetually blanketed by a thick veil of clouds high in carbon dioxide and its surface temperature approaches 900 degrees Fahrenheit.

Launched on Nov. 3, 1973 atop an Atlas-Centaur rocket, Mariner 10 flew by Venus in 1974.

BIRTHS

1608 Caspar Schott SJ, and Gaspar Schott or Kaspar Schott (February 5 1608 in Königshofen, May 22 1666 in Würzburg) was a scientific author and educator.
Schott attended the Würzburg Jesuit High School and entered the Order in 1627. During his studies in Würzburg one of his teachers was Athanasius Kircher. When the Jesuits fled before the approaching Swedish army in 1631,Schott went to Palermo to complete his studies. He stayed in Sicily 20 years as a teacher of mathematics, philosophy, moral theology at the Jesuit school in Palermo. In 1652 was sent to Rome as support in the scientific work of Kircher. He decided to publish Kircher's work. In 1655, he returned as Professor in the Würzburg school, where he taught mathematics and physics. He was Hofmathematker and confessor of the Elector Johann Philipp von Schönborn who had just purchased the vacuum pump invented by Otto von Guericke and used at Magdeburg.
He corresponded with leading scientists including Otto von Guericke, Christiaan Huygens, and Robert Boyle. The term "technology" was probably invented by Schott in his "Technica curiosa" which inspired Boyle and Hooke's vacuum experiments.
In the posthumously published work Organum mathematicum he describes his Cistula invented by him, a computing device, by Kircher with which you can multiply and divide.

The device described in Schott's book was divided by functionality into 9 main sections, each of which contained approximately 24 rods.

Arithmetic The arithmetic rods included a set of Napier's Bones. They were capable of assisting with the multiplication of multi-digit numbers and producing quotients.

Geometry The rods in this section could aid in determining heights, by use of a geometric square.

Fortifications The rods in this section could aid in determining the design of bulwarks in fortification plans.

ChronologynThe rods in this section could be used to determine the date of Easter and other church holidays which were positioned relative to it. These rods simply contained a table of upcoming dates.

Horography The rods in this section contained information needed to construct sundials.

Astronomy This compartment had tablets which resembled those found in an almanac. For each day of the year, the length of the day and night, the times for sunrise and sunset, and the duration of morning and evening twilight were provided. All the information was based on measurements taken at 48 degrees latitude (Vienna).

Astrology This section had tables describing movements for the visible planets, and the constellation Draco, and also provided astrological interpretations for the 12 zodiac signs.

Cryptography The rods in this section could be used to encrypt and decrypt text using a cyclic transposition cypher, based on a keyword.

Music The rods in this section could be used by non musicians to compose church music. The system used was the same as that used for Kircher's previous device, the Arca Musarithmica. They contained sets of musical phrases which could be combined randomly to set verses to music, producing millions of hymns in 4-part harmony.*Wik

the Organum Mathematicum at the Museo Galileo in Florence, Italy.

1797 Jean-Marie-Constant Duhamel (5 Feb 1797; 29 Apr 1872) French mathematician and physicist who proposed a theory dealing with the transmission of heat in crystal structures based on the work of the French mathematicians Jean-Baptiste-Joseph Fourier and Siméon-Denis Poisson. *TIS

1836 Alexander Stewart Herschel (5 February 1836 – 18 June 1907) was a British astronomer, born in Feldhausen, South Africa.
He was the son of John Herschel and the grandson of William Herschel. Although much less well known than either of them, he did pioneering work in meteor spectroscopy. He also worked on identifying comets as the source of meteor showers. The Herschel graph, the smallest non-Hamiltonian polyhedral graph, is named after Herschel due to his pioneering work on Hamilton's Icosian game. *Wik
The image of the graph at right is from Christian Perfect at the Aperiodical Blog. You can’t draw a path on it that visits each vertex exactly once, but you can make a polyhedron whose vertices and edges correspond with the graph exactly. It’s also bipartite – you can color the vertices using two colors so that edges only connect vertices of different colors.
I think the polyhedron is the only enneahedron (9 faces children) that has all quadrilateral faces. You can see the solid here.

Rosalind joined the University of Lausanne in 1917. She also helped her father's research between 1919 and 1921 at the University College Wales in Aberystwyth, and worked with Edward Collingwood, also of Aberystwyth, on a translation of Georges Valiron's course on Integral Functions. She received a L-És-sc (a bachelor's degree) from Lausanne in 1925.

In 1972 she and Ivor Grattan-Guinness published a second edition of her parents' book The Theory of Sets of Points, originally published in 1906.

Rosalind Tanner died on 24 November 1992

R. C. H. Young (left, upper) at the ICM 1932

1907 Wilhelm Magnus (February 5, 1907, Berlin, Germany – October 15, 1990, New York City) made important contributions in combinatorial group theory, Lie algebras, mathematical physics, elliptic functions, and the study of tessellations.*Wik

1915 Robert Hofstadter (5 Feb 1915, 17 Nov 1990) American scientist who was a joint recipient of the Nobel Prize for Physics in 1961 for his investigations in which he measured the sizes of the neutron and proton in the nuclei of atoms. He revealed the hitherto unknown structure of these particles and helped create an identifying order for subatomic particles. He also correctly predicted the existence of hte omega-meson and rho-meson. He also studied controlled nuclear fission. Hofstadter was one of the driving forces behind the creation of the Stanford Linear Accelerator. He also made substantial contributions to gamma ray spectroscopy, leading to the use of radioactive tracers to locate tumors and other disorders.*TIS

1932 Lee Aaron Segel ( 5 February 1932 – 31 January 2005) was an Israeli-American applied mathematician. He developed both the Keller-Segel model of chemotaxis, in cell biology, and the Newell-Whitehead-Segel equation, in fluid dynamics. He also co-authored the first simulation model for herbicide resistance evolution. He is also considered one of the forefathers of the field of theoretical immunology.

Segel was active in the Santa Fe Institute, the first of the over 50 research centers which focus, today, on complex physical, computational, biological, and social systems. Segel was also editor-in-chief of the Bulletin of Mathematical Biology from 1986 to 2001 and co-authored the first volume in the SIAM Classics in Applied Mathematics series, created by the Society for Industrial and Applied Mathematics. He migrated between numerous prestigious academic institutions worldwide, culminating at Israel's Weizmann Institute of Science,where he served as dean of the Faculty of Mathematics and Computer Science and chair of the Scientific Council. *Wik

1930 Kazimierz Urbanik (born 5 February 1930 in Krzemieniec - 29 May 2005 in Wrocław ) - Polish mathematician, rector of the University of Wroclaw ( 1975 - 1981 ), Doctor Honoris Causa of the University of Lodz and the Technical University of Wroclaw. He dealt with problems from different fields of mathematics, but his research interests were focused on the theory of probability . He obtained several important results in the theory of stochastic processes , information theory , theoretical physics , universal algebra , topology and measure theory . He published about 180 scientific papers. *Wik

DEATHS

1881 Thomas Carlyle (4 Dec 1795 in Ecclefechan, Dumfriesshire, Scotland - 5 Feb 1881 in Chelsea, London, England) was a Scottish writer who was also interested in mathematics. He translated Legendre's work.*SAU

1939 Gheorghe Ţiţeica ((October 4, 1873–February 5, 1939) publishing as George or Georges Tzitzeica) was a Romanian mathematician with important contributions in geometry. He is recognized as the founder of the Romanian school of differential geometry.*Wik

1865 Benjamin Franklin Finkel (July 5, 1865 – February 5, 1947) was a mathematician and educator most remembered today as the founder of the American Mathematical Monthly magazine. Born in Fairfield County, Ohio and educated in small country schools, Finkel received both bachelor's and master's degrees from Ohio Northern University, then known as Ohio Normal University (1888 and 1891, respectively).

In 1888 he copyrighted A Mathematical Solution Book. The purpose of the book was to provide mathematics teachers a text utilizing a systematic method of problem solving, "The Step Method", representing a chain of reasoning, in logical order, to arrive at the correct result. The first edition was postponed until 1893, due to financial problems of the original publisher. The book's preface stated that the work was based upon eight years of teaching in the public schools. This was followed by following editions in 1897, 1899 and 1902. [The book treats very basic Arithmetic. *PB]

In 1895 he became professor of mathematics and physics at Drury University, then known as Drury College. He was a University Scholar in Mathematics at the University of Chicago from 1895–1896. In 1906 he was awarded a doctorate from the University of Pennsylvania, where he had earlier earned an additional master's degree in 1904 and a Harrison fellow appointment in 1905.

He was a member of the American Mathematical Society, 1891; the London Mathematical Society, 1898; and Circolo Matematico di Palermo, 1902. He retained his professorship at Drury College until his death in 1947. *Wik

1977 Oskar Benjamin Klein (September 15, 1894 – February 5, 1977) was a Swedish theoretical physicist. Klein retired as professor emeritus in 1962. He was awarded the Max Planck medal in 1959. He is credited for inventing the idea, part of Kaluza–Klein theory, that extra dimensions may be physically real but curled up and very small, an idea essential to string theory / M-theory. *Wik

1980 Nachman Aronszajn (26 July 1907, Warsaw, Poland – 5 February 1980 Corvallis, Oregon, U.S) was a Polish American mathematician of Ashkenazi Jewish descent. Aronszajn's main field of study and expertise was mathematical analysis. He also contributed to mathematical logic.
He received his Ph.D. from the University of Warsaw, in 1930, in Poland. Stefan Mazurkiewicz was his thesis advisor. He also received a Ph.D. from Paris University, in 1935; this time Maurice Fréchet was his thesis advisor. He joined the Oklahoma A&M faculty, but moved to the University of Kansas in 1951 with his colleague Ainsley Diamond after Diamond, a quaker, was fired for refusing to sign a newly-instituted loyalty oath. Aronszajn retired in 1977. He was a Summerfield Distinguished Scholar from 1964 to his death.
He introduced, together with Prom Panitchpakdi, the injective metric spaces under the name of "hyperconvex metric spaces". Together with Kennan T. Smith, Aronszajn offered proof of the Aronszajn–Smith theorem. Also, the existence of Aronszajn trees was proven by Aronszajn; Aronszajn lines, also named after him, are the lexicographic orderings of Aronszajn trees.
He also has a fundamental contribution to the theory of reproducing kernel Hilbert space, the Moore–Aronszajn theorem is named after him. *Wik

1988 Dorothy Lewis Bernstein (April 11, 1914 – February 5, 1988) was an American mathematician known for her work in applied mathematics, statistics, computer programming, and her research on the Laplace transform.
Dorothy Bernstein was born in Chicago, the daughter of Russian immigrants to the US. She was a member of the American Mathematical Society and the first woman elected president of the Mathematical Association of America. Due in great part to Bernstein's ability to get grants from the National Science Foundation, Goucher College (where she taught for decades) was the first women's university to use computers in mathematics instruction in the 1960s.*Wik

1997 Frederick Justin Almgren,(3 July 1933 in Birmingham, Alabama, USA - 5 Feb 1997 in Princeton, USA) Almost certainly Almgren's most impressive and important result was only published in 2000, three years after his death. Why was this? The paper was just too long to be accepted by any journal. Brian Cabell White explains the background in a review of the book published in 2000 containing the result:

By the early 1970s, geometric analysts had made spectacular discoveries about the regularity of mass-minimizing hypersurfaces. (Mass is area counting multiplicity, so that if k sheets of a surface overlap, the overlap region is counted k times.) In particular, the singular set of an m-dimensional mass-minimizing hypersurface was known to have dimension at most m - 7. By contrast, for an m-dimensional mass-minimizing surface of codimension greater than one, the singular set was not even known to have m-measure 0. Around 1974, Almgren started on what would become his most massive project, culminating ten years later in a three-volume, 1700-page preprint containing a proof that the singular set not only has m-dimensional measure 0, but in fact has dimension at most (m - 2). This dimension is optimal, since by an earlier result of H Federer there are examples for which the dimension of the singular set is exactly (m - 2). ...

Now, thanks to the efforts of editors Jean Taylor and Vladimir Scheffer, Almgren's three-volume, 1700-page typed preprint has been published as a single, attractively typeset volume of less than 1000 pages.

Fred Almgren received many honours for his outstanding contributions. He was an Alfred P Sloan Fellow in 1968-70, an Exchange Visitor at the Steklov Mathematical Institute in Leningrad in 1970, a John Simon Guggenheim Memorial Fellow in 1974-75, and Earle Raymond Hedrick Lecturer for the Mathematical Association of America in1975. He was elected a fellow of the American Association for the Advancement of Science in 1982, was awarded a medallion by Brown University in 1988:-

... in recognition of distinguished contributions to society through scholarship and professional activity ...

and he received the Class of 55 Public Service Award from Princeton University in 1988:-

... for contributions to society beyond the bounds of occupation.

Among his service we should mention he was an editor of three journals: the Journal of Experimental Mathematics, the Journal of Geometric Algebra, and Differential Geometry and its Applications. He administered the Geometry Supercomputer Project for the Geometry Computing Group and served on the American Mathematical Society Committee on Applications of Mathematics.*SAU

Photograph by Paul Halmos

2003 Werner Romberg (born 16 May 1909 in Berlin; died 5 February 2003 in Heidelberg) was a German mathematician and physicist.

Romberg studied mathematics and physics form 1928 in Heidelberg and Munich and completed his doctorate in 1933 at Munich University under the supervision of Arnold Sommerfeld; his thesis was entitled "Zur Polarisation des Kanalstrahllichtes" ["On the polarisation of channel light beams"]. In Munich he studied mathematics under, among others, Oskar Perron and Constantin Carathéodory. In 1933, as a so-called "half-Jew" in the terminology of the new National Socialist government of Germany, he sought to emigrate to the Soviet Union. From 1934 to 1937 he worked as a theoretical physicist in the university of Dnipro (then Dnipropetrovsk). In 1938 he went, via the Institute for Astrophysics in Prague, to Norway, where he became an assistant to Egil Hylleraas at the University of Oslo. He also briefly worked at the Technical University of Trondheim with Johan Holtsmark, who was building a Van de Graaff generator there. With the German occupation of Norway he fled to Uppsala in Sweden. In 1941 the Nazi German state stripped him of his German citizenship, and in 1943 recognition of his doctorate was revoked. He became a Norwegian citizen in 1947.

After the Second World War, from 1949 to 1968, he was a professor in Trondheim; from 1960 he was head of the applied mathematics department. In Norway he built up his research group in numerical analysis, and part of the introduction of digital computers, such as GIER, the first computer at Trondheim. From 1968 he held the chair for Mathematical Methods in Natural Sciences and Numerics at Heidelberg University. *Wik