Timothy and Urban are playing a game with two six-sided dice. The dice are unusual: Rather than bearing a number, each face is painted either red or blue.
The two take turns throwing the dice. Timothy wins if the two top faces are the same color, and Urban wins if they’re different. Their chances of winning are equal.
The first die has 5 red faces and 1 blue face. What are the colors on the second die?
Discovered by R.V. Heath in 1950:
Think of two positive integers.
Add them to get a third number.
Add the second number and the third number to get a fourth number.
Continue in this way until you have 10 numbers.
The sum of the 10 numbers is 11 times the seventh number.
Mathematician Yutaka Nishiyama of the Osaka University of Economics has designed a nifty paper boomerang that you can use indoors. A free PDF template (with instructions in 70 languages!) is here.
Hold it vertically, like a paper airplane, and throw it straight ahead at eye level, snapping your wrist as you release it. The greater the spin, the better the performance. It should travel 3-4 meters in a circle and return in 1-2 seconds. Catch it between your palms.
By 1958 many of the attributes of living things could be found in our technology: locomotion (cars), metabolism (steam engines), energy storage (batteries), perception of stimuli (iconoscopes), and nervous or cerebral activity (computers). The missing element was reproduction: We hadn’t yet created a nonliving artifact that could make copies of itself.
So Brooklyn College chemistry professor Homer Jacobson built one. Using an HO gauge model railroad, he designed an “organism” made of boxcars that could use sensors to select other cars on the track and assemble them on a siding into models of itself. “Head” cars have “brains,” and “tail” cars have “muscles” and “eyes”; together, a head and a tail make an organism in which the head directs the tail to watch for available cars elsewhere on the track and shunt them appropriately onto a siding to create a new organism.
“Any new ‘organisms’ formed continue the propagation in a linear fashion,” Jacobson wrote, “until the environment runs out of parts, or there are no more sidings available, or a mistake is made somewhere in the operation of a cycle, i.e., a ‘mutation.’ Such an effect, like that with living beings, is usually fatal.”
(Homer Jacobson, “On Models of Reproduction,” American Scientist, September 1958.)
A conventional balloon rises because its airbag displaces a large volume of air. But the gas that fills the bag has some weight; it, along with the weight of the gondola, reduces the balloon’s total lift.
Realizing this, Italian monk Francesco Lana de Terzi in 1670 proposed a “vacuum airship,” a balloon whose airbag was filled with nothing at all. Since a vacuum weighs nothing, this should maximize the vehicle’s lift — the vacuum could displace a large volume of air without itself adding any weight.
In principle this might work; the problem is that the vacuum would tend to collapse its container, and building a shell sturdy enough to withstand it would leave us with a ship too heavy to lift. It’s not clear whether any material or structure could overcome this problem.
Lee Sallows tells me that the postal system of Macau is releasing a new series of stamps based on magic squares. The full set will touch on everything from the Roman SATOR square to Dürer’s Melencolia. Details are here.
Charmingly, the values of the stamps will be 1, 2, …, 9 Macau patacas, so that the sheet of the nine stamps will itself form a classic Lo Shu magic square. Lee’s contribution, above, is a Nasik 2D geomagic square of order 3 — not only are all the rows and columns magic, but so are all six diagonals, including the four “broken” diagonals.
Somewhat related: In 2000 Finland issued seven stamps in classic tangram shapes, featuring images of science and education. (One of the small triangles, barely visible here, is a Sierpinski gasket.) Only three of the seven shapes are denominated postage, but I should think the temptation is overwhelming to arrange all seven on an envelope in the shape of a little man or a fish or something. I wonder what the post office makes of that.
Two circles intersect. A line AC is drawn through one of the intersection points, B. AC can pivot around point B — what position will maximize its length?
Howard C. Saar of Albion, Mich., pointed out an innovative solution to this problem in Recreational Mathematics Magazine, April 1962:
log(3x + 2) + log(4x – 1) = 2log11
Divide each side of the equation by the word “log”:
(3x + 2) + (4x – 1) = (2)(11)
7x = 21
x = 3
… which is correct.
Claude Shannon, the father of information theory, took an active interest in juggling. He used to juggle balls while riding a unicycle through the halls of Bell Laboratories, and he built the first juggling robot from an Erector set in the 1970s. (The machine above mimics W.C. Fields, who himself juggled in vaudeville before turning to comedy.)
Noting that juggling seems to appeal to mathematics-minded people, Shannon offered the following theorem:
F is flight time, the time the ball spends in the air
D is “dwell time,” the time it spends in the hand
V is vacancy, the time a hand spends empty
B is the number of balls
H is the number of hands
“Theorem 1 allows one to calculate the range of possible periods (time between hand throws) for a given type of uniform juggle and a given time of flight,” he wrote. “A juggler can change this period, while keeping the height of his throws fixed, by increasing dwell time (to increase the period) or reducing dwell time to reduce the period. The total mathematical range available for a given flight time can be obtained by setting D = 0 for minimum range and V = 0 for maximum range in Theorem 1. The ratio of these two extremes is independent of the flight time and dependent only on the number of balls and hands.”
To measure dwell times, Shannon actually created a “jugglometer” in which a juggler wore copper mesh over his fingers and juggled foil-covered lacrosse balls; catching the ball closed a connection between the fingers and started a clock. “Preliminary results from testing a few jugglers indicate that, with ball juggling, vacant time is normally less than dwell time, V ranging in our measurements from fifty to seventy per cent of D.”
Shannon noted that juggling gets dramatically harder as the number of balls increases. He worked out a foolproof solution, at least in theory. A light ray that starts at one focus of an ellipse will be reflected to the other focus. If the ellipse is rotated around its major axis, it will create an egglike shell with two foci. Now if a juggler stands with a hand at each focus, then a ball thrown from either hand, in any direction, will bounce off the shell and arrive at the other hand!
(“Scientific Aspects of Juggling,” in Claude Elwood Shannon: Collected Papers, 1993.)
In 1980 Philip K. Dick was asked to forecast some significant events in the coming years. Among his predictions:
1983: The Soviet Union will develop an operational particle-beam accelerator, making missile attack against that country impossible. At the same time the U.S.S.R. will deploy this weapon as a satellite killer. The U.S. will turn, then, to nerve gas.
1989: The U.S. and the Soviet Union will agree to set up one vast metacomputer as a central source for information available to the entire world; this will be essential due to the huge amount of information coming into existence.
1993: An artificial life form will be created in a lab, probably in the U.S.S.R., thus reducing our interest in locating life forms on other planets.
1997: The first closed-dome colonies will be successfully established on Luna and on Mars. Through DNA modification, quasi-mutant humans will be created who can survive under non-Terran conditions, i.e., alien environments.
1998: The Soviet Union will test a propulsion drive that moves a starship at the velocity of light; a pilot ship will set out for Proxima Centaurus, soon to be followed by an American ship.
2000: An alien virus, brought back by an interplanetary ship, will decimate the population of Earth, but leave the colonies on Luna and Mars intact.
2010: Using tachyons (particles that move backward in time) as a carrier, the Soviet Union will attempt to alter the past with scientific information.
Also: “Computer use by ordinary citizens (already available in 1980) will transform the public from passive viewers of TV into mentally alert, highly trained, information-processing experts.”
(From David Wallechinsky, Amy Wallace, and Irving Wallace, The Book of Predictions, 1980.)
