Mathematicians know that the path from adolescent competence to adult professional life is full of unexpected twists and turns, and is fraught with potential pitfalls. Even when faced with very able adolescents, it is not at all clear which of them will flourish in the long run, or how one can best support their future development (when they eventually become young adults) in a way that takes their mathematical ability and inclinations seriously – whether in pure or applied mathematics, in computer science or software development, in engineering, mathematical biology, financial mathematics or linguistics, in management, or in some totally different profession.

In a prospective audit of consecutive referrals for fetal echocardiography between 1997 and 2003, we documented all instances of an abnormal arrangement of the abdominal vessels. We then established the structure of the heart in these fetuses, noting any extra-cardiac associations, and the eventual outcomes.

We found a cardiac abnormality in 572 of the 2,136 fetuses examined during this period (27 percent), with 16 (0.8 percent) having an abnormal arrangement of the abdominal great vessels. Mirror-imaged arrangement was found in 3, while the arrangement suggested right isomerism in 6, and left isomerism in 7. Of these 16 fetuses, 14 had cardiac malformations. Isomerism of the right atrial appendages was found in 7 fetuses, all with either a right-sided stomach or cardiac apex, and 6 with a common atrioventricular junction guarded by a common valve. Only 1 of these fetuses survived. Of the 3 fetuses with mirror-imaged abdominal great vessels, 2 also had mirror-imaged atrial arrangement, while the 3rd had isomeric right appendages. Only one of 7 fetuses with an abdominal great vein posterior to the aorta had bilateral left atrial appendages. The remaining 6 had usual atrial arrangement, with normal pulmonary venous connections in all but one, who had infra-diaphragmatic totally anomalous pulmonary venous connection. Of these fetuses, 2 had coarctation of the aorta, and 2 others had complex cardiac malformations resulting in neonatal death. One died in childhood from biliary atresia, and three are alive.

Abnormal arrangements of the abdominal great vessels, therefore, were found in 0.8 percent of our total fetuses, and in 2.4 percent of those with cardiac malformations. Those with an abnormal arrangement of the abdominal vessels combined with an abnormal arrangement of the atrial appendages, however, made up only 0.5 percent and 1.6 percent, respectively. While screening for an abnormal arrangement of the abdominal vessels overestimated the incidence of left, but not right, isomerism of the atrial appendages, it did predict the presence of important extra-cardiac malformations that required urgent recognition and management after birth.

The nth triangular number is defined to be the sum of the first n positive integers:

Thus

In a letter to Mersenne in 1638 [1, p. 61], Fermat claimed that every positive integer can be written as a sum of at most three triangular numbers. This remarkable result was eventually proved by Gauss in 1796, at the age of 19.