Conundrums & Reefs

for Dangerously-Little-Knowledge

For each question, just hit on the link-word(s) to find the answer.
[Most of these items are not original with DIO.
*Those that might be (several more probably aren't) are asterisked.]

• Flip a coin four times to test the heads-vs-tails count.
Which is the most likely split: 4-0, 3-1, or 2-2?

• Those who teach physics might enjoy DIO's simple challenge-experiment: blow up a balloon and tell a standing kid that if he releases it aimed upwards, he can't make it hit the ceiling — but you can release it while lying on the floor and still hit the ceiling!

• If you travel from NY to Boston at 40 mph and return at 60 mph, what is your average speed?

• In a group of 30 people, what are the odds that some pair of people will share a one common birthday .

• Anagram-Country: Take the letters in “NEW DOOR” and make one word out of them.

• What are the most northern, western, and eastern states of the US?

• How can you poke your head through a doughnut without destroying it?

• What do you get when you multiply a sphere's diameter times its circumference?

• *How many men have won a US presidential election's popular vote more than twice?

• A smart monkey is hanging from a tree, watching a hunter aim a rifle at him from such a huge distance that the monkey will have time to react before the bullet arrives. But he knows that if he drops before the shot, the hunter will merely re-aim. So he waits for the flash of the muzzle and drops. What happens?

• *What are the middle names of Clarence Thomas, Henry Kissinger, and James Bond?

• What is the shape of the path of a shot cannonball?

• *Is the North Star always within 1° of true north?

• *You're emptyhanded, your sneakered feet right on the edge of a 50 story building's roof, with your center-of-gravity slightly past the edge. Can you save yourself?

• Question: Who was the last person to touch composer Franz Schubert?
Vienna: Schubert's grave on right, Beethoven's on left.
(Monument to Mozart at center. Brahms' grave off-camera to right.)

• What mountain should you climb, in order to stand as high as possible above the Earth's center?

• How arrange 10 trees in 5 rows of 4 trees each?
[Hint: the Jewish version is 12 trees, 6 rows.]

• *Who was first to see the North Pole?

• Go south from Florida's S.tip, KeyWest: which South American nation will you hit?

• What are the odds on two randomly-chosen large numbers sharing a prime?

• *A math oddity DR happens not to have seen cited anywhere (though it has probably been stumbled-upon by numerous persons for centuries):
Add the cubes of all integers from one to N. What are the odds that the result will happen to be an integer squared?

SOLUTIONS to Barbets:

The most likely split is 3-1 — indeed, it's as likely (50%) as the other two put together.
The probability of 2-2 is 3/8. That of 4-0 is 1/8.
[Similar unexpected odds at DIO 13.1 [2003] p.2 n.2 item [a].]

Device invented by DR during Erwin Hiebert's Harvard 1958 Kirkland House physics tutorial, it crudely shows utility of gyroscopes in aiming rockets:
Secret:
Lie back on the floor while blowing up balloon to large size. Then, JUST as you release the balloon upwards from your lips — you violently apply a spin (“couple”) by rotate-whipping both balloon-grasping hands (parallel to the floor) simultaneously. (Hands obviously should move in opposite directions.) The balloon will spiral towards the ceiling and will usually hit it, so long as the house is not too old. (Old houses have high ceilings.) This is a fine combination of: bar-bet, teaching-tool, & show-off.
Of course, you'll only use it to teach.

(This counter-intuitive result is explained at DIO 10 [2000] n.91 [p.44].)

Better than even, since the odds of no common birthday are 365!/335!/365^30 = less than 30%. For any group of 23  or more people, the odds of a match exceed 1/2.

Rearranging the letters of “NEW DOOR” creates the successive letters which spell “ONE WORD”.

All the US' three extreme N-W-E points are in Alaska:
Northernmost: Pt.Barrow.
Westernmost: Amatignak I.
Easternmost: Amchitka I.

The sphere's area.

Poke a finger through the doughnut while poking your head with finger's tip.

Three:
Andrew Jackson: 1824, 1828, 1832.
Grover Cleveland: 1884, 1888, 1892.
Franklin Roosevelt: 1932, 1936, 1940, 1944.
No one won a majority in 1824, and a deal between J.Q.Adams (who became president with a record-low of 31% of the vote) and H.Clay (13%) robbed Jackson (41%) — which helped Jackson win the next two elections with ease (56% & 54%). F.Roosevelt always got over 1/2 the vote. By contrast, Cleveland always got less than 1/2; and, though out-doing his rivals in all 3 of his runs, he lost in the Electoral College in 1888.

While the monkey drops, so does the bullet, exactly the same distance and (down) direction, in the same time (due to the same gravitational acceleration) — and so monkey and bullet have a perfect meet.

Respective middle names of Thomas, Kissinger, and Bond:
Uncle, S.
[are fake middle initials always “S”? — Grant (a rumor?), Truman, & Kissinger (where the S is entirely a DIO invention)]
and James.
[Doesn't 007 ALWAYS superciliously introduce himself as “Bond James Bond”?]

Not a parabola, but the tip-end of an ultra-slim 4000-mile-long ellipse.

Polaris is now only about 0°.7 away from the North Celestial Pole, i.e., a North Polar Distance (NPD) of 0°.7. Nonetheless, for anyplace more than c.45° from the Equator, Polaris is more than a degree from true north twice per day. So the answer to our above question is no for all of Alaska and Greenland, most of Europe & Canada, and (though it hardly matters) all of Tierra del Fuego and Antartica. Problem: the further Polaris is from one's horizon, the further a vertical from it will stray from true north at that horizon. Applying a useful differential approximation: diurnal maximum azimuthal deviation = NPD/cos(latitude).
(DIO 13.1 [2003] ‡1 §H10 & eq.3 [p.10], and Fig.2 [p.11].)

Yes. Violently revolve your outstretched arms — around axes parallel to the edge: left arm clockwise (to you), right arm counter-clockwise.
Just stand on a line or edge (NOT ATOP A 50' HEIGHT! — we're just experimenting here, OK?!) and start falling slightly forward and then try righting yourself without moving your feet: you'll find yourself instinctively doing the foregoing move. It opposes your falling's torque with your arms' opposite (mean) torque, using sneaker-friction to make your feet's contact-point (the building-edge in theory) an axis for your body's life-saving rotation.

Answer: after Schubert's skull had suffered a phrenological exam, symphonic composer Anton Bruckner placed it (1888/9/21) into the Viennese coffin in which Schubert's then-60-yr-old corpse has resided ever since.
(See Viennese native and observer Frederic Morton's learned if creepy history, A Nervous Splendor: Vienna 1888-1889 1979 Chap.9.)

Equatorial-bulge-enhanced Mount Chimborazo of Equador not Mt.Everest is the highest point from Earth's center.
[This entry mistakenly specified Argentina's Mt.Aconcogua, until modestly corrected on 2017/8/25 by Oscar Ferro of Argentina. DIO thanks him for saving us from the potential embarrassment of the author of “Bar-Bets” losing a bar-bet!]

Just plant the trees in the form of a star.

*Wm.Herschel! He saw both the north & south poles of Mars over a century before the Earth's north & south poles were 1st seen: 1926 & 1911, resp (DIO 10 [2000]), by Norway's Roald Amundsen & Oskar Wisting.
Note: for both Mars and Earth, EACH pole was 1st seen by the same men.

None. South America is entirely east of The Key West meridian.

Almost 40%, much larger than seems likely at 1st glance.

Answer: 100%. The result will in fact be the square of the sum of all integers from 1 to N.
[E.g., 1 + 2 + 3 + 4 + 5 = 15; while 1 + 8 + 27 + 64 + 125 = 225 = 15 squared.]