According to Culum Brown from Macquarie University, "Fish are more intelligent than they appear. In many areas, such as memory, their cognitive powers match or exceed those of ‘higher’ vertebrates including non-human primates."
Individual carp captured by anglers have been shown to become less catchable thereafter. This suggests that fish use their memory of negative experiences to associate capture with stress and therefore become less easy to catch.
This type of learning has been shown not just in carp but also in paradise fish (Macropodus opercularis) which avoid places where they have experienced a single attack by a predator and continue to do so for many months. Also, several fish species are capable of learning complex spatial relationships and forming mental maps  and integrate experiences which enable the fish to generate appropriate avoidance responses. This means that a fish can exhibit strong aversive behavior if exposed to injury or a predator. As a result, any reduction in the stressfulness of capture by an angler should be beneficial to angling in the long-term, since recapture of the fish should be less difficult.
Some fish species can exhibit long-term memory. Channel catfish can remember the human voice call announcing food five years after last hearing that call; goldfish remember the colour of a tube dispensing food one year after the last tube presentation; sockeye salmon still react to a light signal that precedes food arrival up to eight months since the last reinforcement; some common rudd and European chub could remember the person who trained them to feed from the hand, even after a 6-month break; crimson-spotted rainbowfish can learn how to escape from a trawl by swimming through a small hole in the center and they remember this technique 11 months later; rainbow trout can be trained to press a bar to get food, and they remember this three months after last seeing the bar; Red Sea clownfish can recognize their mate 30 days after it was experimentally removed from the home anemone.
Tool use is often perceived as a sign of intelligence in animals, but it may be that the behaviours are mostly innate. There are not many examples of tool use in fishes, perhaps because fishes have only their mouth in which to hold objects. Several species of wrasse, and the blackspot tuskfish, have been seen holding clams or urchins in their mouth and smashing them against the surface of a rock (an "anvil") to break them up; whitetail damselfish clean the rock face where they intend to lay eggs by blowing sand grains from their mouth onto it; triggerfish can blow water onto urchins to turn them over and expose their more vulnerable ventral side; archerfish can squirt jets of water at insects that sit on plants above the surface to knock them off their perch and into the water, and they can learn to shoot at moving targets; banded acaras, Bujurquina vittata, lay their eggs on a loose leaf, and they carry the leaf away when a predator approaches.
As for tool use, construction behaviour may be mostly innate but it can be sophisticated in a way that suggests intelligence. Construction methods in fishes can be divided into three categories: excavations, pile-ups, and gluing.
Excavations may be simple depressions dug up in the substrate, such as the nests of bowfin, smallmouth bass, and Pacific salmon, but it can also consist of fairly large burrows used for shelter and for nesting. Burrowing species include the mudskippers, the red band-fish Cepola rubescens (burrows up to 1 m deep, often with a side branch), the yellowhead jawfish Opistognathus aurifrons (chambers up to 22 cm deep, lined with coral fragments to solidify it), the convict blenny Pholidichthys leucotaenia whose burrow is a maze of tunnels and chambers thought to be as much as 6 m long,  and the Nicaragua cichlid, Hypsophrys nicaraguensis, who drills a tunnel by spinning inside of it.. In the case of the mudskippers, the burrows are shaped like a J and can be as much as 2 m deep. Two species, the giant mudskipper Periophthalmodon schlosseri and the walking goby Scartelaos histophorus, build a special chamber at the bottom of their burrows into which they carry mouthfuls of air. Once released the air accumulates at the top of the chamber and forms a reserve from which the fish can breathe – like all amphibious fishes, mudskippers are good air breathers. If researchers experimentally extract air from the special chambers, the fish diligently replenish it. The significance of this behaviour stems from the facts that at high tide, when water covers the mudflats, the fish stay in their burrow to avoid predators, and water inside the confined burrow is often poorly oxygenated. At such times these air-breathing fishes can tap into the air reserve of their special chambers .
Mounds are easy to build, but can be quite extensive. In North American streams, the male cutlip minnow Exoglossum maxillingua, 90-115 mm long (3.5-4.5 in), assembles mounds that are 75-150 mm high (3-6 in), 30-45 cm in diameter (12-18 in), made up of more than 300 pebbles 13-19 mm in diameter (a quarter to half an inch). The fish carry these pebbles one by one in their mouths, sometimes stealing some from the mounds of other males. The females deposit their eggs on the upstream slope of the mounds, and the males cover these eggs with more pebbles. Males of the hornyhead chub Nocomis biguttatus, 90 mm long (3.5 in), and of the river chub Nocomis micropogon, 100 mm long (4 in), also build mounds during the reproductive season. They start by clearing a slight depression in the substrate, which they overfill with up to 10,000 pebbles until the mounds are 60-90 cm (2-3 ft) long (in the direction of the water current), 30-90 cm wide (1-3 ft), and 5-15 cm high (2-6 in). Females lay their eggs among those pebbles. The stone accumulation is free of sand and it exposes the eggs to a good water current that supplies oxygen. Males of many mouthbrooding cichld species in Lake Malawi and Lake Tanganyika build sand cones that are flattened or crater-shaped at the top. Some of these mounds can be 3 m in diameter and 40 cm high. The mounds serve to impress females or to allow species recognition during courtship.
Male pufferfish, Torquigener sp., also build sand mounds to attract females. The mounds, up to 2 m in diameter, are intricate with radiating ridges and valleys.
Several species build up mounds of coral pieces either to protect the entrance to their burrows, as in tilefishes  and gobies of the genus Valenciennea,  or to protect the patch of sand in which they will bury themselves for the night, as in the Jordan's tuskfish Choerodon jordani  and the rockmover wrasse Novaculichthys taeniourus.
Male sticklebacks are well known for their habit of building an enclosed nest made of pieces of vegetation glued together with secretions from their kidneys. Foam nests, made up of air bubbles glued together with mucus from the mouth, are also well known in gouramis and armoured catfish.
Fish can remember the attributes of other individuals, such as their competitive ability or past behavior, and modify their own behavior accordingly. For example, they can remember the identity of individuals to whom they have lost in a fight, and avoid these individuals in the future; or they can recognize territorial neighbors and show less aggression towards them as compared to strangers. They can recognize individuals in whose company they obtained less food in the past and preferentially associate with new partners in the future.
Fish can seem mindful of which individuals have watched them in the past. In an experiment with Siamese fighting fish, two males were made to fight each other while being watched by a female, whom the males could also see. The winner and the loser of the fight were then, separately, given a choice between spending time next to the watching female or to a new female. The winner courted both females equally, but the loser spent more time next to the new female, avoiding the watcher female.  In this species, females prefer males they have seen win a fight over males they have seen losing,  and it therefore makes sense for a male to prefer a female that has never seen him as opposed to a female that has seen him lose.
Knowing that if A>B and B>C, then A>C, is another type of evidence for intelligence, and it can be applied in the context of dominance hierarchies. In a study with the cichlid Astatotilapia burtoni, eight observer fish could watch individual A beat individual B, then B over C, C over D, and D over E. The observer fish were then given a choice of associating with either B or D (both of which they had seen win once and lose once). All eight observer fish spent more time next to D. Fish in this species prefer to associate with more subordinate individuals, so the preference for D showed that the observers had worked out that B was superior to C, and C to D, and therefore D was subordinate to B.
A few examples of deception suggest that fishes can put themselves in the mind of others, though it remains possible that the behaviors are mostly innate. In the threespine stickleback, males sometimes see their nest full of eggs fall prey to groups of marauding females; some of the males, when they see a group of females approaching, move away from their nest and start poking their snout in the ground, as would a female raiding a nest. This commonly fools the females into thinking that a nest has been discovered there and they rush to that site, leaving the male's real nest alone. Bowfin males caring for their free-swimming fry do something similar when a potential fry predator approaches: they move away and thrash about as if injured, drawing the predator's attention onto himself.
In the Malili Lakes of Sulawesi, Indonesia, one species of sailfin silverside, Telmatherina sarasinorum, is an egg predator. It often follows courting pairs of the closely-related species T. antoniae. When those pairs lay eggs, T. sarasinorum darts in and picks at the eggs, eating them. On four different occasions in the field (out of 136 observation bouts in total), the following behaviour was witnessed: a male T. sarasinorum who was following a pair of courting T. antoniae eventually chased off the male T. antoniae and took his place, courting the heterospecific female. That female released eggs, at which point the male fell upon the eggs and ate them. This sneaky courtship behaviour on the male’s part may simply be innate, but it is tempting to interpret it as a deliberate attempt at deception in order to get food.
Cooperative foraging reflects some mental flexibility and could therefore be interpreted as intelligence. There are a few examples in fishes.
Yellowtail amberjack can form packs of 7-15 individuals that maneuver in U-shaped formations to cut away the tail end of prey shoals (jack mackerels or Cortez grunts) and herd the downsized shoal next to seawalls where they proceed to capture the prey.
In the coral reefs of the Red Sea, roving coralgrouper that have spotted a small prey fish hiding in a crevice sometimes visit the sleeping hole of a giant moray and shake their head at the moray, and this seems to be an invitation to group hunting as the moray often swims away with the grouper, is led to the crevice where the prey hides, and proceeds to probe that crevice (which is too small to let the grouper in), either catching the prey by itself or flushing it into the open where the grouper grabs it.
Similarly, zebra lionfish that detect the presence of small prey fishes flare up their fins as an invitation to other zebra lionfish, or even to another species of lionfish (Pterois antennata), to join them in better cornering the prey and taking turns at striking the prey so that every individual hunters ends up with similar capture rates.
Experiments at the University of Padova in Italy have revealed that mosquitofish, Gambusia holbrooki, can distinguish better than chance (but not all the time) between two doors marked with either two or three geometric symbols, only one of which allowed the fish to rejoin its shoalmates. They could do this even when the array of two symbols covered the same total surface area, had the same density, and had the same brightness as the array of three symbols. Additional experiments showed that this discrimination persisted (again above chance level, but only at around 60% accuracy) when the two doors were marked with 4 vs 8, 15 vs 30, 100 vs 200, 7 vs 14, and 8 vs 12 symbols, again controlling for non-numerical factors.
Many studies have shown that when given a choice, shoaling fish prefer to join the largest of two shoals. It has been argued that several aspects of such choice reflect an ability by fish to distinguish between numerical quantities. 
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