Dice

Imagine a world without random numbers. In the 1940s, the generation of statistically random number was important because was important to physicists simulating thermonuclear explosions, and today, many computer networks employ random numbers to select unbiased samples of potential voters.
Dice, originally made  from anklebones of roofed animals, were one of the earliest means for producing random numbers. In ancients civilizations, the gods were believed to control the outcome of dice tosses; thus, dice were relied upon to make crucial decisions, ranging from the selection of rulers to the division of property in an inheritance. even today, the metaphor of God controlling dice  is common, as evidence by astrophysicist Stephen Hawking's quote, "Not only God play dice, but He sometimes confuses us by throwing them where they can't be seen."
the oldest known dice were excavated together with a 5,000 year backgammon set from the legendary Burnt City in southeastern Iran. The city represents four stages of civilization that were destroyed by fires before being abandoned in 2100 B.C. At this same site, archeologists also discovered the earliest known artificial eye, which once stared out hypnotically from the face of an ancient female priestess or soothsayer.
For centuries, dice rolls have been used to teach probability. For a single roll of an n-sided die with a different number for each face, the probability of rolling any value is 1/n. The probability of ruling a particular sequence of i numbers is 1/n^i. For example, the chances of rolling a 1 followed by 4 on a traditional die is 1/6^2 = 1/36. Using two traditional dice, the probability of throwing any given sum is the number of ways to throw that sum divided by the total number of combinations, which is a sum divided by the total number of combinations, which is why a sum of 7 is much more likely than a sum of 2.

Quipu

The ancient incas used quipus (pronounced "key-poos"), memory banks made of strings and knots, for storing numbers. Until recently, the oldest-known quipus dated from about A.D.650. However, in 2005, a quipu from the Peruvian coast city of Caral was dated about 5,000 years ago. The incas of south America had a complex civilization with common state religion and a common language. Although they did not have writing, they kept extensive records encoded by logical numerical system on quipus, which varied in complexity from the three to around three thousand cords. Unfortunately, when the Spanish came to South America, they saw the strange quipus and thought they were works of the Devil. The Spanish destroyed thousands of them in the name of God, and today only 600 quipus remain. Knots types and positions, cord directions, cord levels, and color and spacing represent numbers mapped to real-world objects. Different knot groups were used for different powers of 10. The knots were probably used to record human and material resources and calendar information. The quipus may have contained more information such as construction plans, dance patterns, and even aspects of inca history. The quipu is significant because it dispels the notion that mathematics flourishes only after a civilization has developed writing; however, societies can reach advanced states without ever having developed written records. Interesting, today there are computer systems whose file managers are called quipus, in honor of this very useful ancient device. One sinister application of the quipusby the incas was as a death calculator. Yearly quotas of adults and children were ritually slaughter, and this enterprise was planned using a quipu. Some quipus represented the empire, the cords referred to roads and the knots to sacrificial victims.

Ishango bone

In 1960, Belgium geologist and explorer Jean de Heinzelin de Braucourt (1920-1998) discovered a baboon bone with markings in what is today the Democratic Republic of Congo. The Ishango bone, with it's sequences of notches, was first thought to be a simple tally stick used by a Stone Age African. However, according to some scientists, the marks suggest a mathematical prowess that goes beyond counting of objects. The bone was found in Ishango, near the headwaters of the Nile River, the home of a large population of upper Paleolithic people prior to a volcanic eruption that buried the area. One column of marks on the bone begins with three notches that double to six notches. Four notches double to eight. Ten notches halve to five. This may suggest a simple understanding of doubling and halving. Even more striking is the fact that numbers in other columns are all odd (9, 11, 13, 17, 19, 21). One column contains the prime numbers between 10 and 20, and the numbers in each column sum to 60 or 48, both multiplies 12. A number of tally sticks have been discovered that predate the Ishango bone. For example, the Switzerland Lebombo bone is a 37,000-yard baboon fibula with 29 notches. A 32,000-year old tibia with 57 notches, grouped in fives, was found in Czechoslovakia. Although quite speculative, some have hypnothized that the markings on the Ishango bone form a kind of lunar calendar for a Stone Age woman who kept their menstrual cycles, giving rise to the slogan "menstruation created mathematics." even if the Ishango was a simple bookkeeping device, these tallies seems to sets us apart from animals and represent the first steps to symbolic mathematics. The full mystery of the Ishango bone can't be solved until other similar bones are discovered.

Knots

The use of knots may predate modern humans (Homo Sapiens). For example, seashells colored with ocher, pierced with holes, and dated to 82,000 years ago have been discovered in a Moroccan cave. Other archeological evidence suggests much older bead use in humans. The piercing implies the use of cords and the use of a knot to hold the objects to a loop, such as a necklace.
The quintessence of ornamental knots is exemplified by The Book of Kells, an ornately illustrated Gospel Bible, produced by Celtic monks in about A.D. 800. In modern times, the study of knots, such as the trefoil knot with three crossings, is part of a vast branch of mathematics dealing with closed twisted loops. In 1914, German mathematician Max Denh (1878-1952) showed that the trefoil knot's mirror images are not equivalent.
For centuries, mathematicians ave tried to develop ways to distinguish tangles that look like knots (called unknots) from true knots and to distinguish true knots from one another. Over the years, mathematicians have created seemingly endless tables of distinct knots. So far, more than 1.7 million nonequivalent knots with pictures containing 16 or fewer crossings have been identified.
Entire conferences are devoted to knots today. Scientists study knots in fields ranging from molecular genetics - to help us understand how to unravel a loop of DNA - to particle physics, in an attempt to represent the fundamental nature of elementary particles.
Knots have been crucial to the development of civilization, where they have been used to tie clothing, to secure weapons to the body, to create shelters, and to permit the sailing of ships and world exploration. Today, knot theory in mathematics has become so advanced that mere mortals find it challenging to understand its most profound applications. In a few millennia, humans have transformed knots from simple necklace ties to models of the very fabric of reality.

Cicada-generated prime numbers

Cicadas are winged insects that evolved around 1.8 million years ago during the Pleistocene epoch, when glaciers advanced and retreated across North America. Cicadas of the genus Magicicada spend most of their lives below ground, feeding on the juices of plant roots, and then emerge, mate, and die quickly. These creatures display a startling behavior: Their  emergence is synchronized with periods of years that are usually the prime numbers 13 and 17. (A prime number is an integer such as 11, 13, and 17 that has only two integers divisors: 1 and itself.) During the spring of their 13th or 17th year, these periodical cicadas construct an exit tunnel. Sometimes more than 1.5 million individuals emerge in a single acre; this abundance of bodies may have survival value as they overwhelm predators such as birds that cannot possibly eat them all at once.
Some researchers have speculated that the evolution of prime-number life cycles occurred so that the creatures increased their chances of evading shorter-lived predators and parasites. For example, if the cicadas had a 12-year life cycles, all predators with life cycles of 2, 3, 4, or 6 years might more easily find the insects. Mario Markus of the Max Planck Institute for Molecular Physiology in Dortmund, Germany, and his co-workers discovered that these kind of prime-nimber cycles arise naturally from evolutionary mathematical models of interactions between predator and prey. In order to experiment, they first assigned random life-cycle durations to their computer-simulated populations. After some time, a sequence of mutations always locked the synthetic cicadas into a stable prime-number cycle.
Of course, this research is still in it's infancy and many questions remain. What is special about 13 and 17? What predators or parasites have actually existed to drive the cicadas to these periods? Also, a mystery remains as to why, of the 1,500 cicada species worldwide, only a small number of genus Magicicada are known to be periodical.

Primates count

Around 60 million years ago, small, lemur-like primates had evolved in many areas of the world, and 30 million years ago, primates with monkeylike characteristics existed. Could such creatures count? The meaning of counting by animals is a highly contentious issue among animal behavior experts. However, many scholars suggest that animals have some sense of number. H. kalmus writes in his nature article "Animals as mathematicians": There is now little doubt that some animals such as squirrels or parrots can be trained to count... Counting faculties have been reported in squirrels, rats, and for pollinating insects. Some of these animals and others can distinguish numbers in otherwise similar visual patterns, while others can be trained to recognize and even to reproduce sequences of acoustic signals. A few can even be trained to recognize and even to reproduce sequences of acoustic signals. A few can even be trained to tap out the number of the elements (dots) in a visual pattern... The lack of the spoken numeral and the written symbol makes many people reluctant to accept animals as mathematicians. Rats have been shown to "count" by performing an activity the correct number of times in exchange for a reward. Chimpanzees can press numbers on a computer that match numbers of bananas in a box. Gesture Matsuzawa of the primate research institute at Kyoto university in Japan taught a chimpanzee to identify numbers from 1 to 6 by pressing the appropriate computer key when she was shown a certain number of objects on the computer screen. Michael Beran, a research scientist at Georgia state university in Atlanta, Georgia, trained chimps to use a computer screen and joystick. The screen flashed a numeral and then a series of dots, and the chimps had to match the two. One chimp learned numerals 1 to 7, while another managed to count to 6. When the chimps were tested again after a gap of three years, both chimps were able to match numbers, but with double the error rate.

Ant odometer

Ants are social insects that evolved from vespoid wasps in the mid-Cretaceous period, about 150 million years ago. After the rise of flowering plants, about 100 million years ago, ants diversified into numerous species. The Sahara desert ant travels immense distances over sandy terrain, often completely devoid of landmarks, as it searches for food. These creatures are able to return to their nests using a direct route rather than by retracing their outbound path. Not only they judge directions, using light from the sky for orientation, but they also appear to have a built-in "computer" that functions like a pedometer that counts their steps and allows them to measure exact distances. An ant may travel as far as 160 feet (about 50 meters) until it encounter a dead insect, whereupon it tears a piece to carry directly back to its nest, accessed via a hole often less than a millimeter in diameter. By manipulating the leg lengths of ants to give them longer and shorter strides, a research team of German and Swiss scientists discovered that ants "count" steps to judge distance. For example, after ants had reached their destination, the legs were lengthened by adding stilts or shortened by partial amputation. The researchers then returned the ants so that the ants could start on their journey back to the nest. Ants with the stilts traveled too far and passed best entrance, while those with the amputated legs did not reach it. However, if the ants started their journey from their nest with the modified legs, they were able to compute the appropriate distances. This suggests that stride length is the crucial factor. Moreover, the highly sophisticated computer in the ant's brain enables the ant to compute a quantity related to the horizontal projections of its path so that it doesn't not become lost even if the sandy landscape develops hills and valleys during its journey.