A Population Puzzle

http://commons.wikimedia.org/wiki/File:Chodowiecki_Basedow_Tafel_6_c.jpg

If I have two children, there’s a 50 percent chance that I’ll have a boy and a girl.

But if I have four children, the chance that I’ll have an equal number of boys and girls drops to 6 in 16, or 37.5 percent.

This trend continues — as the number of offspring rises, the chance of having precisely the same number of boys and girls drops:

2 children: 50 percent
4 children: 37.5 percent
8 children: 27.34375 percent
16 children: 19.6380615 percent
32 children: 13.9949934 percent
64 children: 9.9346754 percent
128 children: 7.0386092 percent

This is worrying. Does it mean that in a large population Jacks might drastically outnumber Jills?

No. As the population grows, the distribution assumes the shape of a normal bell curve concentrated near 1/2. So while the chance of precise parity drops, the chance that a large population will have approximately equal numbers of girls and boys actually increases, as we’d expect.

| Science & Math

Both Sides Now

Bach’s “crab canon” rendered as a Möbius strip:

Bach and Handel were both blinded by the same oculist, John Taylor, “the poster child for 18th-century quackery,” according to University of Wisconsin ophthalmologist Daniel Albert. Bach probably died of a post-operative infection; Handel wrote the lyrics to Samson (“Total eclipse! No sun, no moon! / All dark amidst the blaze of noon!”) after Taylor’s botched cataract surgery.

Random Möbius anecdote: In 1957, B.F. Goodrich patented a half-twisted conveyor belt for carrying hot material such as cinders and foundry sand, “thereby permitting each face of the belt to cool during one half of the operating period.”

| Art · Science & Math · Technology

Going Down

http://books.google.com/books?id=CpgkAQAAIAAJ&printsec=frontcover#v=onepage&q&f=false

What would happen if you jumped into a tunnel that passed through the center of the earth? If you encountered no air resistance, dinosaurs, Mad Hatters, or Morlocks, you’d accelerate until you passed through the center at 18,000 mph, then slow as you ascended through the opposite hemisphere. At the far end you’d have just time to tip your hat to the surprised antipodeans before you fell home again, and you’d continue oscillating like this forever.

“If this shaft had its starting-point on one of the mountain plateaux of South America at an elevation of seven thousand feet,” wrote Camille Flammarion in 1909, “and if it issued at the sea-level at the other side, a man who had fallen into the shaft would arrive at the antipodes still travelling at such a speed that the spectators would see this strange projectile shot to a height of seven thousand feet into the air.”

On the other hand, if our straight tunnel connected two points that were not precise antipodes, then we could install a train powered by gravity — it would roll “downhill” on the first part of its journey, and momentum would carry it through the second (again neglecting air resistance and friction). Curiously, in all these cases the total trip would take the same length of time — about 42 minutes.

| Science & Math

Misc

  • Newton was born the year that Galileo died.
  • Cole Porter’s summer home was called No Trespassing.
  • 66339 = (6 × 6)3 + 39
  • Could you have had different parents?
  • “A good conscience is a continual Christmas.” — Ben Franklin

UPDATE: The first item here is incorrect. The dates coincide only if one uses the Gregorian calendar to date Galileo’s death and the Julian to date Newton’s birth. The two events occurred 361 days apart, which puts them in separate years on both calendars. Apparently this is a very common error. (Thanks, Igor.)

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Snappy New Year

In Insurmountable Simplicities (2006), Roberto Casati points out that a traveler flying east may miss midnight — by entering a new time zone, he may jump from the 11:00 hour into the 12:00 hour without passing through midnight:

Mathematics and geography tell us that during the flight it will happen more than once that we reach the stroke of the hour without leaving the time zone we are in. If our flight lasts eight hours and the time difference between New York and Paris is six hours, this will happen at least twice, and at most eight times. But there is no guarantee that the stroke of midnight will be among these cases.

Thus, if it’s New Year’s Eve, an eastbound traveler may get no champagne.

| Science & Math

Frozen Fire

https://commons.wikimedia.org/wiki/File:PSM_V20_D677_Fulgurite_tubes_created_by_vitrification_of_sand_by_lighting.jpg

Lightning can fuse sand into curious rootlike tubes up to 5 meters long, called fulgurites. Because their shape records the path of the strike as it passes into the ground, they’re sometimes known as petrified lightning.

Lightning had a ruinous history before the introduction of Ben Franklin’s lightning rod. The campanile of St. Mark in Venice was destroyed three times over. In 1769, a bolt struck the tower of St. Nazaire in Brescia, whose magazine contained 100 tons of gunpowder. One-sixth of the town was destroyed, and 3,000 people died.

Compounding the harm was the disastrous belief that ringing bells during thunderstorms would allay lightning. In one 33-year period, lightning struck 386 church towers and killed 103 bell ringers.

Modern strikes are less dire. In 1919, Cleveland Indians pitcher Ray Caldwell was struck by lightning during a game against the Philadelphia Athletics. “It felt like a sandbag hit me,” he said. He refused to leave the game and pitched to Joe Dugan for the final out. The Indians won, 2-1.

| Science & Math · Technology

Crowded

Any set of 10 positive integers smaller than 100 will always contains two subsets with the same sum.

In any such group, the number of possible subsets (excluding the empty set) is 210 – 1, or 1023. And the largest possible sum of any subset is 90 + 91 + … + 99 = 945. Hence, no matter which numbers are chosen, there will always be more subsets than possible sums, and some subsets (dozens, actually) must yield the same sum.

| Science & Math