Bon Appétit
- It is now true that Clarence will have a cheese omelette for breakfast tomorrow. [Premise]
- It is impossible that God should at any time believe what is false, or fail to believe anything that is true. [Premise: divine omniscience]
- Therefore, God has always believed that Clarence will have a cheese omelette for breakfast tomorrow. [From 1, 2]
- If God has always believed a certain thing, it is not in anyone’s power to bring it about that God has not always believed that thing. [Premise: the unalterability of the past]
- Therefore, it is not in Clarence’s power to bring it about that God has not always believed that he would have a cheese omelette for breakfast. [From 3, 4]
- It is not possible for it to be true both that God has always believed that Clarence would have a cheese omelette for breakfast, and that he does not in fact have one. [From 2]
- Therefore, it is not in Clarence’s power to refrain from having a cheese omelette for breakfast tomorrow. [From 5, 6]
So Clarence’s eating the omelette tomorrow is not an act of free choice.
From William Hasker, God, Time, and Knowledge, quoted in W. Jay Wood, God, 2011.
The Ballot Box Problem
In 1878 W. A. Whitworth imagined an election between two candidates. A receives m votes, B receives n votes, and A wins (m>n). If the ballots are cast one at a time, what is the probability that A will lead throughout the voting?
The answer, it turns out, is given by the pleasingly simple formula
Howard Grossman offered the proof above in 1946. We start at O, where no votes have been cast. Each vote for A moves us one point east and each vote for B moves us one point north until we arrive at E, the final count, (m, n). If A is to lead throughout the contest, then our path must steer consistently east of the diagonal line OD, which represents a tie score. Any path that starts by going north, through (0,1), must cut OD on its way to E.
If any path does touch OD, let it be at C. The group of such paths can be paired off as p and q, reflections of each other in the line OD that meet at C and continue on a common track to E.
This means that the total number of paths that touch OD is twice the number of paths p that start their journey to E by going north. Now, the first segment of any path might be up to m units east or up to n units north, so the proportion of paths that start by going north is n/(m + n), and twice this number is 2n/(m + n). The complementary probability — the probability of a path not touching OD — is (m – n)/(m + n).
(It’s interesting to consider what this means. If m = 2n then p = 1/3 — even if A receives twice as many votes as B, it’s still twice as likely that B ties him at some point as that A leads throughout.)
Theory and Practice
In 1982 Richard Feynman and his friend Tom Van Sant met in Geneva and decided to visit the physics lab at CERN. “There was a giant machine that was going to be rolled into the line of the particle accelerator,” Van Sant remembered later. “The machine was maybe the size of a two-story building, on tracks, with lights and bulbs and dials and scaffolds all around, with men climbing all over it.
“Feynman said, ‘What experiment is this?’
“The director said, ‘Why, this is an experiment to test the charge-change something-or-other under such-and-such circumstances.’ But he stopped suddenly, and he said, ‘I forgot! This is your theory of charge-change, Dr. Feynman! This is an experiment to demonstrate, if we can, your theory of 15 years ago, called so-and-so.’ He was a little embarrassed at having forgotten it.
“Feynman looked at this big machine, and he said, ‘How much does this cost?’ The man said, ‘Thirty-seven million dollars,’ or whatever it was.
“And Feynman said, ‘You don’t trust me?’”
(Quoted in Christopher Sykes, No Ordinary Genius, 1994.)
Pick’s Theorem
Georg Alexander Pick found a useful way to determine the area of a simple polygon with integer coordinates. If i is the number of lattice points in the interior and b is the number of lattice points on the boundary, then the area is given by
There are 40 lattice points in the interior of the figure above and 12 on the boundary, so its area is 40 + 12/2 – 1 = 45.
(Thanks, Pål.)
Misc
- Only humans are allergic to poison ivy.
- GUNPOWDERY BLACKSMITH uses 20 different letters.
- New York City has no Wal-Marts.
- (5/8)2 + 3/8 = (3/8)2 + 5/8
- “Ignorance of one’s misfortunes is clear gain.” — Euripides
For any four consecutive Fibonacci numbers a, b, c, and d, ad and 2bc form the legs of a Pythagorean triangle and cd – ab is the hypotenuse.
(Thanks, Katie.)
Some Palindromes
In the minuet in Haydn’s Symphony No. 47, the orchestra plays the same passage forward, then backward.
When Will Shortz challenged listeners to submit word-level palindromes to National Public Radio’s Weekend Edition Sunday in 1997, Roxanne Abrams offered the poignant Good little student does plan future, but future plan does student little good.
And Connecticut’s Oxoboxo River offers a four-way palindrome — it reads the same forward and backward both on the page and in a mirror placed horizontally above it.
Heart and Soul
From a point P, drop perpendiculars to the sides of a surrounding triangle. This defines three points; connect those to make a new triangle and drop perpendiculars to its sides. If you continue in this way, the fourth triangle will be similar to the original one.
In 1947, Mary Pedoe memorialized this fact with a poem:
Begin with any point called P
(That all-too-common name for points),
Whence, on three-sided ABC
We drop, to make right-angled joints,
Three several plumb-lines, whence ’tis clear
A new triangle should appear.
A ghostly Phoenix on its nest
Brooding a chick among the ashes,
ABC bears within its breast
A younger ABC (with dashes):
A figure destined, not to burn,
But to be dropped on in its turn.
By going through these motions thrice
We fashion two triangles more,
And call them ABC (dashed twice)
And thrice bedashed, but now we score
A chick indeed! Cry gully, gully!
(One moment! I’ll explain more fully.)
The fourth triangle ABC,
Though decadently small in size,
Presents a form that perfectly
Resembles, e’en to casual eyes
Its first progenitor. They are
In strict proportion similar.
The property generalizes: Not only is the third “pedal triangle” of a triangle similar to the original triangle, but the nth “pedal n-gon” of an n-gon is similar to the original n-gon.
Who’s Counting?
In the 14th century, an unnamed Kabbalistic scholar declared that the universe contains 301,655,722 angels.
In 1939, English astrophysicist Sir Arthur Eddington calculated that it contains 15,747,724, 136,275,002,577,605,653,961,181,555,468,044,717,914,527,116,709,366,231,425,076,
185,631,031,296 electrons.
“Some like to understand what they believe in,” wrote Stanislaw Lec. “Others like to believe in what they understand.”
Good Behavior
Back in 2010 I posted a prime magic square created by a prison inmate and published anonymously in the Journal of Recreational Mathematics. The same prisoner composed the 7×7 square above, which has some remarkable properties of its own:
- Here again every cell is prime.
- The numbers in each row, column, and the two main diagonals add to the magic constant of 27627.
- That same constant, 27627, is the sum of each broken diagonal (that is, each pair of parallel diagonals that include seven numbers, for example 3881 + 827 + 9257 + 5471 + 1741 + 29 + 6421).
- If the units digit is removed from each number (changing 9341 to 934, 6367 to 636, etc.), then it remains a pandiagonal magic square, with all the properties mentioned above for the primes.
Both squares appeared in the October 1961 issue of Recreational Mathematics Magazine — editor Joseph S. Madachy noted that they had been “sent to Francis L. Miksa of Aurora, Illinois from an inmate in prison who, obviously, must remain nameless.”
It’s not clear to me why the prisoner shouldn’t get credit for this work, whatever his crime — presumably he created both squares while working alone and without tools or references, a remarkable achievement. If I learn any more I’ll post it here.
Insight
Many a man floated in water before Archimedes; apples fell from trees as long ago as the Garden of Eden, and the onrush of steam against resistance could have been noted at any time since the discovery of fire and its use under a covered pot of water. In all these cases it was eons before the significance of these events was perceived. Obviously a chance discovery involves both the phenomenon to be observed and the appropriate, intelligent observer.
– Walter Cannon, The Way of an Investigator, 1945
Brothers in Binary
A number is said to be perfect if it equals the sum of its divisors: 6 is divisible by 1, 2, and 3, and 1 + 2 + 3 = 6.
St. Augustine wrote, “Six is a number perfect in itself, and not because God created all things in six days; rather the converse is true; God created all things in six days because this number is perfect, and it would have been perfect even if the work of the six days did not exist.”
Perfect numbers are rare. No one knows whether an infinite quantity exist, and no one knows whether any of them are odd. The early Greeks knew the first four, and in the ensuing two millennia we’ve uncovered only 44 more. But they have one thing in common — they reveal a curious harmony when expressed in base 2:
Unquote
“We have not the reverent feeling for the rainbow that a savage has, because we know how it is made. We have lost as much as we gained by prying into that matter.” — Mark Twain
“At last I fell fast asleep on the grass & awoke with a chorus of birds singing around me, & squirrels running up the trees & some Woodpeckers laughing, & it was as pleasant a rural scene as ever I saw, & I did not care one penny how any of the beasts or birds had been formed.” — Charles Darwin, letter to his wife, April 28, 1858
Greetings
Launched in 1976, NASA’s Laser Geodynamic Satellite needed a stable orbit to permit precise measurements of continental drift, so its designers gave it a high trajectory and a heart of solid brass. As a result, it’s not expected to return to Earth for 8 million years. That raised an interesting challenge: What message could we attach to the satellite that might be intelligible to our descendants or successors, who might recover it thousands of millennia in the future?
Tasked with that problem, Carl Sagan came up with the “greeting card” at left, which is affixed to LAGEOS on a small metal plaque. Using it, whoever comes upon the plaque can calculate roughly the time between his own epoch and ours. In Sagan’s words, the card says, “A few hundred million years ago the continents were all together, as in the top drawing. At the time LAGEOS was launched the map of the Earth looks as in the middle drawing. Eight million years from now, when LAGEOS should return to Earth, we figure the continents will appear as in the bottom drawing. Yours truly.”
Balance
For any rectangle, the sum of the squares of the distances from any point P to two opposite corners is equal to the sum of the squares of the distances from that point to the two other corners (so, above, a2 + c2 = b2 + d2). This remains true whether the point is inside or outside the rectangle, on a side or a corner, or even outside the plane.
Pushkin wrote, “Inspiration is needed in geometry, just as much as in poetry.”
Math Notes
10989 × 9 = 98901 × 1
21978 × 8 = 87912 × 2
32967 × 7 = 76923 × 3
43956 × 6 = 65934 × 4
54945 × 5 = 54945 × 5
Correlation, Causation
From Richard F. Mould’s Introductory Medical Statistics — this graph plots the population of Oldenburg, Germany, at the end of each year 1930-1936 against the number of storks observed in that year.
Does this explain the storks’ presence? Not necessarily: In 1888 J.J. Sprenger noted, “In Oldenburg there is a curious theory that the autumnal gatherings of the storks are in reality Freemasons’ meetings.”
Digit Work
A useful system of finger reckoning from the Middle Ages:
To multiply 6 x 9, hold up one finger, to represent the difference between the 5 fingers on that hand and the first number to be multiplied, 6.
On the other hand, hold up four fingers, the difference between 5 and 9.
Now add the number of extended fingers on each hand to get the first digit of the answer (1 + 4 = 5), and multiply the number of closed fingers on each hand to get the second (4 × 1 = 4). This gives the answer, 54.
In this way one can multiply numbers between 6 and 9 while knowing the multiplication table only up to 5 × 5.
A similar system could be used to multiply numbers between 10 and 15. To multiply 14 by 12, extend 4 fingers on one hand and 2 on the other. Add them to get 6; add 10 times that sum to 100, giving 160; and then add the product of the extended fingers, 4 × 2, to get 168.
This system reflects the fact that xy = 10 [(x - 10) + (y - 10)] + 100 + (x – 10)(y – 10).
(From J.T. Rogers, The Story of Mathematics, 1968.)
Misc
- The first child to be vaccinated in Russia was named Vaccinov.
- Every treasurer of the United States since 1949 has been a woman.
- 15642 = 1 + 56 + 42
- up inverted is dn.
- “Life well spent is long.” — Leonardo
Stormy Weather
Take an ordinary magic square and imagine that the number in each cell denotes its altitude above some common underlying plane. And now suppose that it begins to rain, with an equal amount of water falling onto each cell. What happens? In the square at left, the water cascades from square 25 down to square 21, and thence down to 10, 7, 2, and into space; because there are no “lowlands” on this landscape, no water is retained. (Water flows orthogonally, not diagonally, and it pours freely over the edges of the square.)
By contrast, in the square on the right a “pond” forms that contains 69 cubic units of water — as it happens, the largest possible pond on a 5×5 square.
With the aid of computers, these imaginary landscapes can be “terraformed” into surprisingly detailed shapes. Craig Knecht, who proposed this area of study in 2007, created this 25×25 square in 2012:
Next year will mark the 500th anniversary of the famously fertile magic square in Albrecht Dürer’s 1514 engraving Melancholia — a fact that Knecht has commemorated in the shape of the ponds on the 14×14 square at right.
Partial Credit
On his 36th birthday, feeling that his most fertile years were behind him, mathematician Abram Besicovitch said, “I have had four-fifths of my life.”
At age 59 he was elected to the Rouse Ball Chair of Mathematics at Cambridge.
When J.C. Burkill reminded him of his earlier remark, he said, “Numerator was correct.”
Finding Yourself
Twinkle, twinkle, little star,
How I wonder what you are.
Up above the world so high,
Like a diamond in the sky.
Twinkle, twinkle, little star,
How I wonder what you are.
Choose any word in the first two lines, count its letters, and count forward that number of words. For example, if you choose STAR, which has four letters, you’d count ahead four words, beginning with HOW, to reach WHAT. Count the number of letters in that word and count ahead as before. Continue until you can’t go any further. You’ll always land on YOU in the last line.
See Finding Religion and The Kruskal Count.
Soulmates
In 1966, asked to describe the person least likely to develop atherosclerosis, Cambridge research fellow Alan N. Howard answered, “A hypotensive, bicycling, unemployed, hypo-beta-lipoproteinic, hyper-alpha-lipoproteinic, non-smoking, hypolipaemic, underweight, premenopausal female dwarf living in a crowded room on the island of Crete before 1925 and subsisting on a diet of uncoated cereals, safflower oil, and water.”
Oxford physician Alan Norton added that her male counterpart was an ectomorphic Bantu who worked as a London bus conductor, had spent the war in a Norwegian prison camp, never ate refined sugar, never drank coffee, always ate five or more small meals a day, and was taking large doses of estrogen to check the growth of his prostate cancer.
“All these phrases mark correlations established in the last few years in a field of medical research which, in volume at least, is unsurpassed,” noted Richard Mould in Mould’s Medical Anecdotes. “The conflict of evidence is unequalled as well.”
Ho
If this sentence is true, then Santa Claus exists.
If that sentence is true, then it’s the case that Santa Claus exists. But wait — in making this observation, we seem to have confirmed the truth of the original sentence. And if that sentence is true, then Santa Claus exists! Where is the error?
(By Raymond Smullyan.)
Head or Tail
Cihan Altay’s Rotator typeface presents the digits 0-9 whether it’s right side up or upside down.
So this equation:
… can be inverted to make this one:
Both are valid.
(Thanks, Lorenzo.)
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