Opinion Counting on counting

You could trust me to complicate the issue. I assigned variables to the time required and the volume of the tub,formed an equation based on rates of discharge and then solved it mentally,noting happily that the volume variable cancelled itself out. One hour,I announced correctly. But my classmate then showed me how I could have solved the puzzle arithmetically rather than fallen into the algebraic trap he had laid for me. Every minute,the tap fills a 20th part of the tub while the outlet drains out a 30th part; the tub retains 1/20 x 1/30 = 1/60 of itself. A 60th part every minute means a full tub in 60 minutes. It’s as simple as that.

The reason I have dug this up is in part to show how convolutedly the minds of science students worked then — or at least how mine did — and in part to observe that today’s schools can potentially make better mathematicians of our children. Yet they can be groomed even better — by being taught,for instance,IPL as a mathematics chapter in Class VIII,maybe IX.

School effectively encouraged us to express anything we did not know as “x” and form an equation to solve. My daughter,in contrast,has been so groomed as to start many of those same problems with a known — and work her way from there. The Class VIII textbook no longer enforces algebra where arithmetic will do.

This is what my college teachers would have done with the IPL: they would have taught us to treat much of the new format with advanced permutation/combination formulae. But we can teach schoolchildren that arithmetic alone can solve most questions about the IPL — such as the number of matches in a group. Let’s say we introduce a school class to five teams in a hypothetical,single round-robin before taking them to the actual format.

“How many matches does each team play?” the teacher asks. “Four,” the class replies in chorus.

“How many does Team A play?”

“Four,” they repeat.

“And how many other matches does Team B play?” Half the class says “four”; the other half has been put on guard the moment the teacher said “other”. Then the brightest of the silent ones works it out. “Three other matches,” she says,“because we had already counted A vs B among A’s matches.”

“Good,” says Teacher,and continues,“and how many other matches does Team C play?”

“Two other matches,” says the entire class,having got the drift,“because we had already counted two of C’s matches,first A vs C and then B vs C.”

If that doesn’t teach a child that the total number of matches in a round-robin of five teams is 4 + 3 + 2 + 1 = 10,nothing ever will.

This,incidentally,presents a case for teaching children the formula for the sum of consecutive numbers starting with 1: take the last number,multiply by one more than itself,and halve the product. We were taught the derivation first and the application later,a nonsensical sequence of events we must reverse for our children.

To get back to the IPL,it involved 2 double round-robins,with the format complicated further by each team playing 4 teams in the opposite group once and the fifth team (chosen at random) twice. Yet,your child can be taught to keep it simple.

Every team played 8 matches within its group,2 more with one team outside,and 1 each with the other four teams,a total of 14. For 10 teams playing 14 matches each,multiply 14 × 10 = 140,a product that includes every match twice (once in the quota of either team playing that match). The actual total matches are,therefore,140/2 = 70.

Take the same reasoning to the original single round-robin of 5 teams. We have 5 teams playing 4 matches each,a product of 20,which again includes every match twice,so that the actual total is (4 × 5)/2 = 10. In other words,it’s (number of matches) × (1 more)/2,and we are back at the formula for the sum of consecutive numbers. Will it intrigue a child? I am sure it will.

Going back to the bathtub puzzle,my daughter struggled to solve it independently,found my equation horribly complicated and liked the commonsense solution. And yes,she is interested in the IPL,though we haven’t got round to discussing its arithmetic yet.

kabir.firaque@expressindia.com