The Punnet square is a bookkeeping tool that allows us to work out all possible compositions of male or female gametes for a particular cross. For the sake of simplicity, we will refer to these male and female gametes as sperm and egg, respectively. The specific form of the "particle" that is passed on to the offspring is called the "allele." The boxes in the Punnet square help us to keep track of all possible offspring that can be produced by this particular cross.

As with Mendel's experiment on tall versus short pea plants, let's start with the characteristic of "plant height." Each individual has two "particles," called alleles, that correspond to height. One allele was inherited from each parent. "T" represents the allele that confers tallness and "t" represents the allele that is related to short plant height. Recall that Mendel started with pure breeding lines of tall and short plants. Thus, all individuals of our "tall" line have only "T" alleles, and all of the individuals of our "short" line have "t" alleles. The allelic combinations ("genotypes") of the tall and short individuals are "TT" and "tt," respectively. Individuals or genotypes that have two copies of the same allele are called "homozygotes."

In this demonstration, we will cross a tall male with a short female. (We would get the same results if we crossed a tall female with a short male. However, in more complicated crosses, the direction of the cross, with regard to sex, can make a difference.) During fertilization, each parent contributes only one allele for plant height-even though each parent has two alleles, which can be the same or different. In our example, both alleles from the male parent are the same (T's, written as "TT"). Similarly, both alleles that could be contributed by the female are the same (t's, written as "tt"). Here, we are writing the potential male contributions on the top and potential female contributions on the left-hand side. By putting one "T" in each sperm, the alleles are following Mendel's first law of inheritance: The Law of Segregation. The two alleles that made the male tall are now segregating to produce reproductive cells (gametes) that can combine with contributions from the female to form the next generation. The same is true for the female.

Each offspring will receive one allele for plant height from each parent (and thus will end up with two alleles for this characteristic). Write each type of allele that the male can contribute in each box, going down. In this case, the male parent only can contribute a "T" to the gametes that will go to each of the offspring.

Since one "T" ends up in each male gamete, the alleles are following Mendel's first law of inheritance: The Law of Segregation. The two alleles that made the male tall are now segregating to produce the next generation.

Now write in each type of allele that the female can contribute in each box going across. In this case, every offspring gets a "t," again following Mendel's Law of Segregation. Each combination in the grid results in an offspring that is "Tt." A genotype with two different alleles (forms of a characteristic) is called a "heterozygote."

In pea plants, the "T" allele masks the effects of the "t" allele. The terms "dominant" and "recessive" are used for the masking and the covered allele, respectively. All offspring from this cross are heterozygotes in terms of their genotypes. They also are tall (because the allele for tall masks the allele for short) in terms of their "phenotype" (the observable feature produced by the genotype). These alleles thus follow Mendel's third law of inheritance: The Law of Dominance. Mendel had the  advantage of the simple inheritance pattern of pea plant height. Unlike human height, which is determined by multiple factors, plant height is determined by one major heritable factor. In addition, dominant alleles do not always cover up the recessive allele entirely. Sometimes, the dominant allele only partially masks the recessive allele. This is called "incomplete dominance."

Next, write in the female's allelic contributions to the potential offspring. In this case, there are three possible genotypes, in the ratio 1:2:1, for the TT, Tt, and tt combinations, respectively (note that Tt is the same as tT). Since the T allele is dominant to the t allele, every individual with at least one T has the tall phenotype. Only individuals homozygous for "t" will be short. Thus, we find that the phenotypes of the offspring are in the ratio of 3:1 for tall and short, respectively.

Note how this particulate model of inheritance maintains variation. Under the blending model of inheritance, one would expect a mating of two tall individuals to produce only tall offspring. Instead, the crossing of the F₁ recovered the variation in the parental lines.

Increasing now to two traits, we will keep the designations of "T" and "t" for tall and short pea plants, respectively, and then add "R" and "r" for round and wrinkled seeds, respectively. In this example, we will cross a male that is heterozygous for tallness and rounded seeds with a female that has the same genotype (TtRr). A Punnet square will, again, help us to work out all possible genotypes (and phenotypes) of the resultant offspring.

The male can produce four types of sperm, each with one allele for each trait. The four types of sperm for a double-heterozygote male are TR, Tr, tR, and tr.

Because the allele for one gene (or characteristic) is independent of (does not dictate) the allele of the other gene, the alleles are said to assort independently. The is Mendel's second law of inheritance: The Law of Independent Assortment. This law would be violated, for example, if the "T" allele was always associated with the "R" allele (and thus eliminating the possibility of Tr and tR sperm).