Punnett Square Calculator

What a Punnett square calculator does

A Punnett square calculator takes two parents’ genotypes, such as AaBb and AaBb, lists the gametes each can make, and fills a grid with every combination of them, so that each box is one equally likely offspring and counting boxes gives the probability of every genotype and phenotype. For AaBb × AaBb with complete dominance that is 16 boxes, nine genotypes and the phenotype ratio 9:3:3:1. The same probabilities come from multiplying gene by gene, P(AaBb) = P(Aa) × P(Bb), which is why the square and the product rule always agree.

The grid takes its name from the Cambridge geneticist Reginald Punnett. His book Mendelism, in its 1911 edition, calls it the method “sometimes termed the ‘chessboard’ method”: each parent’s series of gametes is written into the squares themselves, one across every row and the other down every column, so that each square pairs one gamete from each parent and “all the possible combinations are represented and in their proper proportions”. The calculator draws the layout textbooks use today instead, with each parent’s gametes written once, along the top or down the side.

How to use it

Type each parent’s genotype with one pair of letters per gene, a capital for the dominant allele and a small letter for the recessive one: Aa is heterozygous, AA and aa are homozygous, and AaBbCc is heterozygous at three genes. Both parents need the same genes, in any order. For each gene choose complete dominance, where AA and Aa look alike, or incomplete dominance or codominance, where the heterozygote has a phenotype of its own. Naming the traits is optional and changes only the labels, and the example buttons load textbook crosses, among them the ones this page works through.

The calculator lists each parent’s gametes, draws the square with parent 1’s gametes across the top and parent 2’s down the side, colours each box by phenotype when there are no more than eight phenotypes, and counts the genotype and phenotype classes as whole-number ratios, fractions and exact percentages. Choose any offspring genotype, from the list or by picking its box, to see its probability worked out by the product rule and checked against the count of boxes. When a cross has from two to 16 phenotypes, a link under the phenotype table opens the chi-square calculator with that ratio already entered, ready for counts of real offspring.

Worked example: Mendel’s round and yellow peas

In his 1866 paper Gregor Mendel crossed peas with round, yellow seeds and peas with wrinkled, green ones, then raised the hybrids and let them self-fertilise. He wrote round as A and wrinkled as a, yellow as B and green as b, so the hybrids were AaBb and the cross is AaBb × AaBb, the calculator’s default. The Mendel’s peas button names his traits.

Each parent is heterozygous at two genes, so it makes 2 × 2 = 4 kinds of gamete, AB, Ab, aB and ab, each 1/4 of the time. The square is 4 × 4 = 16 boxes, each 1/16. Counting them gives nine genotypes in the ratio 1:2:1:2:4:2:1:2:1 and four phenotypes: 9 round yellow, 3 round green, 3 wrinkled yellow and 1 wrinkled green in every 16, which is 56.25%, 18.75%, 18.75% and 6.25%. By the product rule the round yellow share is P(A_) × P(B_) = 3/4 × 3/4 = 9/16, and the double recessive is P(aa) × P(bb) = 1/4 × 1/4 = 1/16.

Mendel counted 556 seeds from this cross: 315 round yellow, 108 round green, 101 wrinkled yellow and 32 wrinkled green. The 9:3:3:1 ratio predicts 312.75, 104.25, 104.25 and 34.75, and the chi-square statistic comes to 0.47 on 3 degrees of freedom, far below the 7.815 needed to reject the ratio at the 5 percent level.

His single-gene counts sit as close. Hybrids for seed shape alone, Aa × Aa, gave him 5474 round and 1850 wrinkled seeds, the ratio 2.96:1 against the 3:1 the square predicts, which the percent error calculator puts at 1.4% short. The spread of repeated counts from one sample to the next is what the standard deviation calculator measures.

Why multiplying works: the product rule

The law of segregation says a parent passes on one allele of each gene, each with probability 1/2, and the law of independent assortment says the allele a gamete takes at one gene tells you nothing about the allele it takes at another. Both follow from how chromosomes separate at meiosis. Together they make each gene a separate event, so the chance of an offspring is the product of its chances gene by gene. From AaBbCc × AaBbCc an aabbcc offspring has probability 1/4 × 1/4 × 1/4 = 1/64, and one with all three dominant traits 3/4 × 3/4 × 3/4 = 27/64, exactly what counting the 64 boxes of the 8 by 8 square gives.

Mendel wrote the counting rule down himself: when two parent stocks differ in n characters, their hybrids’ offspring fall into 3ⁿ classes among every 4ⁿ individuals. So a cross between parents heterozygous at every gene gives 9 genotypes in 16 boxes for two genes and 27 in 64 for three. With complete dominance those genotypes show 2ⁿ phenotypes, in the 3:1 ratio expanded once per gene: 9:3:3:1 for two genes and 27:9:9:9:3:3:3:1 for three. Past three genes the square becomes unwieldy, 256 boxes for four, but the product rule does not: the chance of all four dominant phenotypes is (3/4)⁴ = 81/256.

Incomplete dominance and codominance

Dominance describes how the genotypes look, not how they are passed on, so switching a gene from complete to incomplete dominance changes the phenotypes and leaves the square itself untouched. In snapdragons, red crossed with white gives pink, and pink crossed with pink gives red, pink and white in the ratio 1:2:1, which is the genotype ratio itself; the Snapdragon colour button loads that cross. Codominance gives the same 1:2:1, with the heterozygote showing both parental forms rather than a blend, as in the MN blood group, where a person with one M and one N allele expresses both equally.

The two can be mixed in one cross. With gene A completely dominant and gene B incompletely dominant, AaBb × AaBb gives six phenotypes in the ratio 3:6:3:1:2:1. The letters keep their capital and small forms even where neither allele is dominant; which allele gets the capital is then only a label.

From letters to DNA

Each letter stands for an allele, one version of a gene’s DNA sequence, and the two letters of a pair sit at the same place on a pair of chromosomes. The sickle-cell allele of the haemoglobin gene HBB differs from the usual one by a single base, GAG to GTG in one codon of the coding strand, which turns one glutamic acid in the protein into a valine; the DNA to protein translator reads both codons. The DNA double helix explorer shows the molecule the sequence is written in.

What this does not cover

The square assumes genes on autosomes rather than on the sex chromosomes, genes that assort independently, two alleles per gene, gametes that are all equally likely to form and to be fertilised, and offspring that are all equally likely to survive. Each assumption fails somewhere.

• Linked genes. Genes close together on one chromosome are inherited together more often than chance allows, so their gametes are not equally likely. Bateson, Saunders and Punnett found this in 1905 for flower colour and pollen shape in sweet peas, the “coupling” of Punnett’s Mendelism. The counts then miss the ratio, and a chi-square test, which the chi-square calculator runs, is how the miss shows.
• Sex-linked genes. A gene on the X chromosome, such as white eye colour in fruit flies, gives different ratios in sons and daughters, which a square of autosomal letters cannot show.
• Multiple alleles. The ABO blood groups have three alleles and rabbit coat colour has four. This calculator takes two alleles per gene.
• Epistasis. When one gene masks another, the genotypes here are still right but the phenotypes group differently. Coat colour in mice gives 9:3:4 from AaCc × AaCc, because the 3 A_cc boxes and the 1 aacc box are all albino.
• Lethal combinations. The Curly allele of fruit flies kills its homozygotes, so a cross of two heterozygotes leaves survivors in the ratio 2:1 rather than 1:2:1.
• Populations. A square is one cross between two known parents. How common an allele is across a population is the question the Hardy-Weinberg calculator answers.

Common mistakes

• Putting both alleles of a gene in one gamete. A gamete carries one allele of every gene, so AaBb makes AB, Ab, aB and ab, never Aa or Bb.
• Giving a homozygous parent an allele it lacks. AABb makes only AB and Ab. A small a in any of its gametes is an allele it does not carry.
• Mixing up the two ratios. Aa × Aa is 1:2:1 in genotype and 3:1 in phenotype. For this cross the two match only under incomplete dominance or codominance, where the heterozygote has a phenotype of its own.
• Adding where you should multiply. Across genes, chances multiply. Within one gene, the routes to the same phenotype add: P(A_) = P(AA) + P(Aa) = 1/4 + 1/2 = 3/4.
• Reading a ratio as a promise. 3:1 gives each offspring a 3/4 chance of the dominant trait; it does not say how many in a family will have it. Among families of four children from Aa × Aa parents, only 27/64, about 42%, have exactly three with the dominant trait.
• Taking dominant to mean common. Dominance is about which phenotype a heterozygote shows. How often an allele turns up in a population is a separate question.