452 = 2025
20 + 25 = 45
453 = 91125
9 + 11 + 25 = 45
454 = 4100625
4 + 10 + 06 + 25 = 45
455 = 184528125
18 + 4 + 5 + 2 + 8 + 1 + 2 + 5 = 45
456 = 8303765625
8 + 3 + 0 + 3 + 7 + 6 + 5 + 6 + 2 + 5 = 45
(Thanks, Dheeraj.)
Science & Math
Benham’s Top
Cut out this disc, pierce it with a pencil, and spin it like a top. The colors that appear are not entirely understood; it’s thought that they arise due to the different rates of stimulation of color receptors in the retina. The effect was discovered by the French monk Benedict Prévost in 1826, and then rediscovered 12 times, most famously by the toy maker Charles E. Benham, who marketed an “artificial spectrum top” in 1894. Nature remarked on it that November: “If the direction of rotation is reversed, the order of these tints is also reversed. The cause of these appearances does not appear to have been exactly worked out.”
Misc
- Dorothy Parker left her entire estate to Martin Luther King Jr.
- SOUTH CAMBRIDGE, NY contains 16 different letters.
- 45927 = ((4 + 5) × 9)2 × 7
- STONE AGE = STAGE ONE
- “You cannot be both fashionable and first-rate.” — Logan Pearsall Smith
Upstanding
You can distinguish a raw egg from a hard-boiled one by spinning it.
The reason for this was puzzled out only in 2002 by mathematicians Keith Moffat of Cambridge University and Yatuka Shimomura of Keio University. Friction between the egg and the table produces a gyroscopic effect, and the egg trades some kinetic energy for potential energy, raising its center of gravity. The raw egg can’t do this because its runny interior lags behind the shell. Moffat wrote:
Place a hard-boiled egg on a table,
And spin it as fast as you’re able;
It will stand on one end
With vectorial blend
Of precession and spin that’s quite stable.
Math Notes
Right and Wrong
Can objects have preferences? The rattleback is a top that seems to prefer spinning in a certain direction — when spun clockwise, this one arrests its motion, shakes itself peevishly, and then sweeps grandly counterclockwise as if forgiving an insult.
There’s no trick here — the reversal arises due to a coupling of instabilities in the top’s other axes of rotation — but prehistoric peoples have attributed it to magic.
See Right Side Up.
Some Coincidences
There are almost exactly 500 million inches in the pole-to-pole diameter of the earth.
The speed of light is within 0.1% of 300,000 kilometers per second.
A cube with a side of 1 mile has nearly the same volume as a sphere with a radius of 1 kilometer.
See Applied Math.
(Thanks, Ben.)
High Hopes
A worm crawls along an elastic band that’s 1 meter long. It starts at one end and covers 1 centimeter per minute. Unfortunately, at the end of each minute the band is instantly and uniformly stretched by an additional meter. Heroically, the worm keeps its grip and continues crawling. Will it ever reach the far end?
Heads and Tails
Let’s play a coin-flipping game. At stake is half the money in my pocket. If the coin comes up heads, you pay me that amount; if it comes up tails, I pay you.
Initially this looks like a bad deal for me. If the coin is fair, then on average we should expect equal numbers of heads and tails, and I’ll lose money steadily. Suppose I start with $100. If we flip heads and then tails, my bankroll will rise to $150 but then drop to $75. If we flip tails and then heads, then it will drop to $50 and then rise to $75. Either way, I’ve lost a quarter of my money after the first two flips.
Strangely, though, the game is fair: In the long run my winnings will exactly offset my losses. How can this be?
Rules of Thumb
“If an elderly but distinguished scientist says that something is possible he is almost certainly right, but if he says that it is impossible he is very probably wrong.” — Arthur C. Clarke
“When, however, the lay public rallies around an idea that is denounced by distinguished but elderly scientists and supports that idea with great fervor and emotion — the distinguished but elderly scientists are then, after all, probably right.” — Isaac Asimov