The pessimists said, but wait a moment. These are aliens. Technical and scientific discoveries throughout human history didn’t come in the most convenient or logical order. Archimedes was unlucky. He had the integral calculus within his grasp, and if Arabic numeral notation had been available to him he would have beaten Newton and Leibniz by almost two millennia. Kepler, on the other hand, had been fortunate. The Greeks, from Euclid to Apollonius, had established hundreds of theorems concerning conic sections. When Kepler needed them in order to replace the old systems of epicycles with his own laws, those theorems sat waiting.

Aliens are likely to know different things, because there is no fixed order for discovery. Maybe we have as much to offer them as they have to offer us. Suppose they never invented the alphabet, or positional notation in mathematics? Then their messages could be all ideographs, their numbers Roman numerals. But far more likely they would use something less familiar and comprehensible than either.

Milly had long ago made her own decision as to where she stood. You could not afford to be either an extreme pessimist or an extreme optimist. On the side of pessimism, surely any aliens would be physically and mentally nothing like humans. They were, after all, aliens. Their languages, notations, and order of evolution of ideas would be vastly different. On the other hand, on the side of optimism, surely any alien thought processes must follow the universal laws of logic. Also, anyone who bothered to send messages far across space would want their messages to be not only received, but comprehended.

Once you accepted those two assumptions, you had certain guarantees. To take one simple example, no sensible alien would ever send as part of a message 2 ? 2 = 4 unless there was other independent evidence as to how the symbol ? was to be interpreted. The message was too ambiguous. The receiver could not determine whether ? stood for plus (2 + 2 = 4), times (2X2 = 4), or raise to a power (22 = 4).

If it were up to Milly, she knew exactly how she would build and send a SETI message. First, you defined special symbols that provided start and stop instructions; then you displayed the positive integers, with enough examples, such as sequences of primes, to make sure the receiver could be absolutely sure there was no misinterpretation.

After that came the symbols of common arithmetic, with examples showing how to add, subtract, multiply, and divide. From there it was a short step to negative numbers, fractions, powers, and irrationals. Imaginaries you would introduce using fractional powers of negative numbers. Then on to series of powers, and the elementary transcendental functions such as sines, cosines, logarithms, and exponents. In every case you would give enough examples to be sure there was no confusion. After providing series expressions for the universal transcendentals such as ? or e, you would provide a check that all was being interpreted correctly by quoting one of mathematics’ enduring wonders, a formula that mysteriously links transcendental and imaginary numbers with the basic numerical building blocks of 1 and 0:

e i? + 1=0.

Mathematics was easy, the obvious way to start. After that, Milly would proceed to astronomy, physics, chemistry, and, finally and most difficult, language.

The trouble, of course, was that it was not up to Milly. She was not sending a message. She was receiving one. The difference, in terms of self-confidence, was the difference between being a doctor and being a patient.

The good news was that she was not working alone. People as smart as she, and probably a whole lot smarter, were her allies. The displays in front of her provided an overview of the whole signal in schematic form, subdivided into twenty-seven regions.

Using her console to control the rate at which she advanced, Milly set out to scan the entire length of the signal. The Puzzle Network team had worked cooperatively to attach their analyses to the appropriate regions. The result was like a gigantic snake, of which the string of digits of the signal itself formed a narrow backbone. Here and there, in places where something particularly interesting and significant had been discovered, the snake bulged out like a python that had swallowed a pig.

Milly backed the scan to study Section 7, the fourth bulge, which at first sight was bigger than all the others. The comments were offered in ordered bunches:

Attoboy: The structure here is odd. High entropy sequences of average length 106 digits are regularly interspersed with low entropy regions each of constant length 3.3554 X 107 digits. Any thoughts?

Sneak Attack: Yes. We could be seeing sections of “text” (variable but roughly equal lengths) that introduce or describe a “picture” (something in image format, with a constant array size). Maybe square arrays of black and white images, each about 6,000 X 6,000 elements?

Claudius: More likely, a gray scale image 4,096 X 4,096 (212 X 212 — that supports the notion of binary representations), with 2 bits (4 levels) for each pixel. That fits with the exact size of the low entropy regions, 33,554,432 bits.

Sneak Attack: Could just as well be 2,048 X 2,048, with 256 (8-bit) gray levels.

Claudius: Should be easy enough to find out which. If we assume a particular line length and do cross-correlations of successive lines, the correct line length should jump right out at us when we get to it, because the correlation will be a lot higher. Let me take a look.

That was all for that cluster of messages. Presumably Claudius did not yet have an answer, or at least not one that “jumped right out.” Milly moved on.

The seventh bulge along the signal’s spine, Section 12, contained remarks similar to the previous one, except for three added comments:

Megachirops: In this case the low entropy regions have a constant length of 4,194,304 bits, exactly one-eighth as long as in Section 7. Does anyone else find this somewhat surprising?

Ghost Boy: We would probably make them all the same sire. The difference may be part of the message, trying to tell us something.

Claudius: Or could these be line drawings? — binary images, black and white with no shades of gray.

The ninth bulge supported a hypothesis offered early in the history of SETI:

The Joker: My frequency analysis of this section suggests that we are dealing with base 4 arithmetic, rather than the base 2 binary we have seen elsewhere. The temptation to interpret this as a biological description in terms of strings of four nucleotides is strong.

Attoboy: Beware of anthropomorphism. But I agree, the temptation is strong. I will try to correlate this section with everything in the genome library.

Not surprisingly, Attoboy had not yet reported results from that effort. The task was a monstrous one. The library to be examined held complete genomes for more than two million species, everything from humans and oak trees and mushrooms to the smallest and simplest viruses. No one, no matter how optimistic, would hope for an exact match. It would be a miracle (and enormously relevant to the universal nature of life) if anything correlated at all with a living creature from Earth. But Attoboy was right, you couldn’t afford not to look.

Milly worked her way on through the signal, section by section. The exercise was giving her a strong inferiority complex. The results that she was seeing had been performed so quickly, and offered such powerful evidence of ingenuity — what could she possibly contribute? The team had already established the existence of unique start and stop sequences, each fourteen bits long. Numerical base and reading order were known beyond doubt: integers were base 2 and base 4, with the most significant digit to the right. Sequences of primes and squares and cubes had been discovered, more than long enough to be unambiguous.