How can six people be organized into four committees so that each committee has three members, each person belongs to two committees, and no two committees have more than one person in common?
It’s possible to work this out laboriously, but it yields immediately to a geometric insight:
If each line represents a committee and each intersection is a person, then the problem is solved.
Lyssophobia is fear of hydrophobia.
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.
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.)
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.)
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.)
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.)
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.
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.
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.”
