For the octopus, with thousands of suckers studding symmetric arms, each of which can bend at any point, building a central mental representation of how to move seems like a computational nightmare. But experiments show that the octopus doesn’t do that. “The brain doesn’t have to know how to move this floppy arm,” Cheng said. Rather, the arm knows how to move the arm.

Readings of electric signals show that when a sucker finds a piece of food, it sends a wave of muscle activation inward up the arm. At the same time, the base of the arm sends another wave of clenched muscles outward, down the arm. Where the two signals meet each other, the arm makes an elbow—a joint in exactly the right place to reach the mouth.

Yet another related strategy, this one perhaps much more common and less controversial, is that the sensory systems of many animals are tuned in to the parts of the world that are relevant to their lives. Bees, for example, use ultraviolet vision to find flowers that have also evolved ultraviolet markings. That avoids the need to take in lots of data and parse it later. “If you do not have those receptors, that part of the world simply doesn’t exist,” said William Wcislo, a behaviorist at the Smithsonian Tropical Research Institute in Panama.

And then there are animals that appear to offload part of their mental apparatus to structures outside of the neural system entirely. Female crickets, for example, orient themselves toward the calls of the loudest males. They pick up the sound using ears on each of the knees of their two front legs. These ears are connected to one another through a tracheal tube. Sound waves come in to both ears and then pass through the tube before interfering with one another in each ear. The system is set up so that the ear closest to the source of the sound will vibrate most strongly.

In crickets, the information processing—the job of finding and identifying the direction that the loudest sound is coming from—appears to take place in the physical structures of the ears and tracheal tube, not inside the brain. Once these structures have finished processing the information, it gets passed to the neural system, which tells the legs to turn the cricket in the right direction.

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Extended cognition may partly be an evolutionary response to an outsized challenge. According to a rule first observed by the Swiss naturalist Albrecht von Haller in 1762, smaller creatures almost always devote a larger portion of their body weight to their brains, which require more calories to fuel than other types of tissue.

Haller’s rule holds across the animal kingdom. It works for mammals from whales and elephants down to mice; for salamanders; and across the many species of ants, bees and nematodes. And in this latter range, as brains demand more and more resources from the tiny creatures that host them, scientists like Wcislo and his colleague William Eberhard, also at the Smithsonian, think new evolutionary tricks should arise.