The programming language Chef, devised by David Morgan-Mar, is designed to make programs look like cooking recipes. Variables are represented by “ingredients,” input comes from the “refrigerator,” output is sent to “baking dishes,” and so on. The language’s design principles state that “program recipes should not only generate valid output, but be easy to prepare and delicious,” but many of them fall short of that goal — one program for soufflé correctly prints the words “Hello world!”, but the recipe requires 32 zucchinis, 101 eggs, and 111 cups of oil to be combined in a bowl and served to a single person. Mike Worth set out to write a working program that could also be read as an actual recipe. Here’s what he came up with:

Hello World Cake with Chocolate sauce.

This prints hello world, while being tastier than Hello World Souffle. The main
chef makes a " world!" cake, which he puts in the baking dish. When he gets the
sous chef to make the "Hello" chocolate sauce, it gets put into the baking dish
and then the whole thing is printed when he refrigerates the sauce. When
actually cooking, I'm interpreting the chocolate sauce baking dish to be
separate from the cake one and Liquify to mean either melt or blend depending on

33 g chocolate chips
100 g butter
54 ml double cream
2 pinches baking powder
114 g sugar
111 ml beaten eggs
119 g flour
32 g cocoa powder
0 g cake mixture

Cooking time: 25 minutes.

Pre-heat oven to 180 degrees Celsius.

Put chocolate chips into the mixing bowl.
Put butter into the mixing bowl.
Put sugar into the mixing bowl.
Put beaten eggs into the mixing bowl.
Put flour into the mixing bowl.
Put baking powder into the mixing bowl.
Put cocoa  powder into the mixing bowl.
Stir the mixing bowl for 1 minute.
Combine double cream into the mixing bowl.
Stir the mixing bowl for 4 minutes.
Liquify the contents of the mixing bowl.
Pour contents of the mixing bowl into the baking dish.
bake the cake mixture.
Wait until baked.
Serve with chocolate sauce.

chocolate sauce.

111 g sugar
108 ml hot water
108 ml heated double cream
101 g dark chocolate
72 g milk chocolate

Clean the mixing bowl.
Put sugar into the mixing bowl.
Put hot water into the mixing bowl.
Put heated double cream into the mixing bowl.
dissolve the sugar.
agitate the sugar until dissolved.
Liquify the dark chocolate.
Put dark chocolate into the mixing bowl.
Liquify the milk chocolate.
Put milk chocolate into the mixing bowl.
Liquify contents of the mixing bowl.
Pour contents of the mixing bowl into the baking dish.
Refrigerate for 1 hour.

Worth confirmed that this correctly prints the words “Hello world!”, and then he used the same instructions to bake a real cake. “It was surprisingly well received,” he writes. “The cake was slightly dry (although nowhere near as dry as cheap supermarket cakes), but this was complimented well by the sauce. My brother even asked me for the recipe!”

While we’re at it: Fibonacci Numbers With Caramel Sauce.

Sweet Story


In 1987, paleontologist Tom Rich was leading a dig at Dinosaur Cove southwest of Melbourne when student Helen Wilson asked him what reward she’d get if she found a dinosaur jaw. He said he’d give her a kilo (2.2 pounds) of chocolate. She did, and he did.

Encouraged, the students asked Rich what they’d get if they found a mammal bone. These are fairly rare among dinosaur fossils in Australia, so Rich rashly promised a cubic meter of chocolate — 35 cubic feet, or about a ton.

The cove was “dug out” by 1994, and paleontologists shut down the dig. Rich sent a curious unclassified bone, perhaps a turtle humerus, to two colleagues, who recognized it as belonging to an early echidna, or spiny anteater — a mammal.

Rich now owed the students $10,000 worth of chocolate. “It turns out that it is technically impossible to make a cubic meter of chocolate, because the center would never solidify,” he told National Geographic in 2005. So he arranged for a local Cadbury factory to make a cubic meter of cocoa butter, and then turned the students loose in a room full of chocolate bars.

“It was a bit like Willy Wonka,” Wilson said. “There were chocolate bars on the counters, the tables. We carried out boxes and boxes of chocolate.”

Fittingly, the new echidna was named Kryoryctes cadburyi.

Wheels of Justice


In 1962 a Swedish motorist was fined for leaving his car too long in a space with a posted time limit. The motorist objected, saying that he had removed the car in time and then happened to return to the same spot later, resetting the time limit. The policeman defended his charge, saying that he had noted the positions of the valves on two of the tires — the front-wheel valve was in the 1 o’clock position, the rear-wheel valve at 8 o’clock. If the car had been moved, he argued, the valves were unlikely to take the same positions.

The court accepted the motorist’s claim, calculating that the chance that the valves would return to the same positions by chance was 1/12 × 1/12 = 1/144, great enough to establish reasonable doubt. The court added that if all four valves had been found to be in the same position, the lower likelihood (1/12 × 1/12 × 1/12 × 1/12 = 1/20,736, it figured) would have been enough to uphold the fine.

Is this right? In evaluating this reasoning, University of Chicago law professor Hans Zeisel notes that this method is biased in favor of the defendant, since the positions of the valves are not perfectly independent. He later added, “The use of the 1/144 figure for the probability of the constable’s observations on the assumption that the defendant had driven away also can be questioned. Not only may the rotations of the tires on different axles be correlated, but the figure overlooks the observation that the car was in the same parking spot. When a person leaves a parking place, it is far from certain that the spot will be available later and that the person will choose it again. For this reason, it has been said that the probability of a coincidence is even smaller than a probability involving only the valves.”

(Hans Zeisel, “Dr. Spock and the Case of the Vanishing Women Jurors,” University of Chicago Law Review, 37:1 [Autumn 1969], 1-18)

Shadow Play

Image: Wikimedia Commons

I am watching a double solar eclipse. The heavenly body Far, traveling east, passes before the sun. Beneath it passes the smaller body Near, traveling west. Far and Near appear to be the same size from my vantage point. Which do I see?

Common sense says that I see Near, since it’s closer. But Washington University philosopher Roy Sorensen argues that in fact I see Far. Near’s existence has no effect on the pattern of light that reaches my eyes. It’s not a cause of what I’m seeing; the view would be the same without it. (Imagine, for example, that Far were much larger and Near was lost in its shadow.)

“When objects are back-lit and are seen by virtue of their silhouettes, the principles of occlusion are reversed,” Sorensen concludes. “In back-lit conditions, I can hide a small suitcase by placing a large suitcase behind it.”

See In the Dark.

(Roy Sorensen, “Seeing Intersecting Eclipses,” Journal of Philosophy XCVI, 1 (1999): 25-49.)

World View

In Other Inquisitions, Borges writes of a strange taxonomy in an ancient Chinese encyclopedia:

On those remote pages it is written that animals are divided into (a) those that belong to the Emperor, (b) embalmed ones, (c) those that are trained, (d) suckling pigs, (e) mermaids, (f) fabulous ones, (g), stray dogs, (h) those that are included in this classification, (i) those that tremble as if they were mad, (j) innumerable ones, (k) those drawn with a very fine camel’s hair brush, (l) others, (m) those that have just broken a flower vase, (n) those that resemble flies from a distance.

This is fanciful, but it has the ring of truth — different cultures can classify the world in surprisingly different ways. In traditional Dyirbal, an aboriginal language of Australia, each noun must be preceded by a variant of one of four words that classify all objects in the universe:

  • bayi: men, kangaroos, possums, bats, most snakes, most fishes, some birds, most insects, the moon, storms, rainbows, boomerangs, some spears, etc.
  • balan: women, bandicoots, dogs, platypus, echidna, some snakes, some fishes, most birds, fireflies, scorpions, crickets, the hairy mary grub, anything connected with water or fire, sun and stars, shields, some spears, some trees, etc.
  • balam: all edible fruit and the plants that bear them, tubers, ferns, honey, cigarettes, wine, cake
  • bala: parts of the body, meat, bees, wind, yamsticks, some spears, most trees, grass, mud, stones, noises and language, etc.

“The fact is that people around the world categorize things in ways that both boggle the Western mind and stump Western linguists and anthropologists,” writes UC-Berkeley linguist George Lakoff in Women, Fire, and Dangerous Things (1987). “More often than not, the linguist or anthropologist just throws up his hands and resorts to giving a list — a list that one would not be surprised to find in the writings of Borges.”

In a Word

adj. producing fever

The 1895 meeting of the Association of American Physicians saw a sobering report: Abraham Jacobi presented the case of a young man whose temperature had reached 149 degrees.

Nonsense, objected William Henry Welch. Such an observation was impossible. He recalled a similar report in the Journal of the American Medical Association (March 31, 1891) in which a Dr. Galbraith of Omaha had found a temperature of 171 degrees in a young woman.

“I do not undertake to explain in what way deception was practised, but there is no doubt in my mind that there was deception,” he said. “Such temperatures as those recorded in Dr. Galbraith’s and Dr. Jacobi’s cases are far above the temperature of heat rigor of mammalian muscle, and are destructive of the life of animal cells.”

Jacobi defended himself: Perhaps medicine simply hadn’t developed a theory to account for such things. But another physician told Welch that Galbraith’s case at least had a perfectly satisfactory explanation — another doctor had caught her in “the old-fashioned trick of heating the thermometer by a hot bottle in the bed.”

Self-Tiling Tile Sets

These tiles have a remarkable property — by working together, the four can impersonate any one of their number (click to enlarge):

sallows rep-tiles 1

The larger versions could then perform the same trick, and so on. Here’s another set:

sallows rep-tiles 2

In this set, each of the six pieces is paved by some four of them:

sallows rep-tiles 3

By the fathomlessly imaginative Lee Sallows. There’s more in his article “More on Self-Tiling Tile Sets” in last month’s issue of Mathematics Magazine.


Here’s a way to visualize multiplication that reduces it to simple counting:

multiplication lattice

Express the digits in each factor with rows of parallel lines, as shown, and then count the intersections to derive the product. This is more cumbersome than the traditional method, but its visual nature is appealing, and it permits anyone who can count to reach the right answer even if he doesn’t know the multiplication table.

The example above uses small digits, so no “carrying” is required, but the method does accommodate more complex sums — it’s explained well in this video:

See Two by Two.

(Thanks, Dieter.)


A card trick by Mark Wilson:

wilson card trick

Put your finger on any red card. Move it left or right to the nearest black card. Move it up or down to the nearest red card. Move it diagonally to the nearest black card. Now move it down or to the right to the nearest red card.

You’ll always land on the ace of diamonds.

(From Harold R. Jacobs, Mathematics: A Human Endeavor, 1970.)