The rhyme of reason & logic

There is the story of a door-to-door salesman who encountered a housewife who had studied advanced mathematics. Having inspected the wares the salesman had brought she threw a challenge at him: “Solve my problem and I will buy things from you.” The salesman asked her to state the problem: “It refers to my three daughters,” she replied. “The sum of their ages is 13 and the product of their ages is the same as the number of this house. Tell me their ages.” The salesman noted the house number in his small pocket book and started calculating. But not for long! He came back to the housewife with a complaint: “Madam, you have given me insufficient information. Please tell me some additional but relevant detail so that I may solve your problem.” The lady thought for a while and then came back with an apology. “Sir, I agree that I need to give you more information. Here it is: my eldest daughter plays the harmonica very well.” On hearing this the salesman’s eyes lit up and he responded: “Madam, now I know the answer.” He gave it and the housewife agreed with his solution and bought some items from him. So what was the answer

To the uninitiated this may sound strange. What has harmonica playing to do with the solution The answer illustrates the role of logical reasoning in solving the problem. For example, you might also be worried that, whereas the salesman knew the house number, we don’t so how are we expected to solve the problem

To proceed logically towards the solution, we first partition the total 13 into three parts, in all possible ways; each partition giving three ages which we multiply out. Thus a typical partition three, four, six leads to a product of the ages as 72. If we had partitioned the number differently, say as four, four, five we would have got the product as 80. What the salesman did was to try all the partitions, working out in each case their products, which he compared with the house number. Thus, had the house number been 72, the salesman would have come back with the answer three, four, six as the ages of her daughters. But this obviously did not happen; the salesman asked for further information. Why If you make a list of all partitions you will find why. The house number 36 is the only one that corresponds to two different partitions! They are two, two, nine and also one, six, six. Thus, both qualify for an answer and the salesman needed further information to decide between the two. The clue given by the housewife can now be appreciated: the eldest daughter exists only in the first set, the second set has two elder girls, both aged six and so there is no single eldest daughter. So here is our answer. The three girls are aged two, two and nine with the eldest certainly able to play harmonica!

Note that we have arrived at the correct answer not by any “trick”, but by applying rules of logic that we commonly adhere to.

This subservience to rules of logic is important to argue one’s case in mathematics or in any scientific argument. The internal consistency of any discussion, in fact, demands that this be done. However, care is needed to ensure that the logical sequence is not violated. Else, we might end up with absurd results.

The following example is of some interest. Three persons are walking in single file. One says: “I have two persons ahead of me.” “So have I,” says the second and the third one says: “I too, likewise.” How is it possible Is there a breakdown of logic Not necessarily! They are walking in a circle.

Geometry, which we study at school, owes its origin to the Greek mathematician Euclid. It is a subject that follows logical reasoning and can lead to fallacies, that is nonsensical results if there is something wrong with the way the figures are drawn. For example, when introducing the subject of geometry, Euclid made sure that its basic assumptions (axioms) are clearly stated. They are assumed to be true and any theorem to be proved takes that for granted. Thus, there are well-known fallacies wherein obviously “wrong” results are proved using all the respectable axioms of Euclid. For example, one can “prove” that the sides of any triangle drawn are equal!

A theorem proved in school geometry tells us that the three angles of every triangle add up to 180°. Yet here is a triangle drawn right on our Earth that defies that rule. To construct it, start at the North Pole and label it as point A. Next, walk down south along the Greenwich Meridian to the Equator. Call this point B. Then turn left and walk eastward along the Equator for a quarter of the way and call this point C. From C walk all the way North till you reach the North Pole. You will find that you have returned to your starting point A along a direction making a right angle with your starting direction. This triangle ABC has a right angle at each of its three vertices so that their sum is 270°. What has gone wrong Finding the answer led mathematicians to new geometries.

Apart from the development of mathematics, logical reasoning plays an important role in many other fields, even non-mathematical ones! The Cambridge mathematician and astronomer Arthur Stanley Eddington once set a cricket problem that he thought of while watching cricket at the university’s Fenner’s cricket ground. It gives the scorecard of a batting side with bowling analysis of the bowling side. There are a few simplifying assumptions, like all runs were scored in singles and fours. But given this information you are asked to give a ball-by-ball description of the entire match. The solution to the problem demonstrates the remarkable power of logical reasoning.

The writer, a renowned astrophysicist, is professor emeritus at Inter-University Centre for Astronomy and Astrophysics, Pune University Campus. He was Cambridge University’s Senior Wrangler in Maths in 1959.