Math Games in an Educational Context (Part 1 of 3)

As a pupil at an English prep school in the 1970s, the curriculum was surprisingly predictable. English, Maths, Science, History and Geography were interspersed with languages, both ancient and modern, and a weekly period of divinity to keep us on the straight and narrow. However, the first year pupils – those aged seven and eight – were timetabled to have one lesson each week with the headmaster. In modern parlance, these periods would be described as ‘logic and verbal reasoning’, but in reality they comprised a strange mixture of riddles, brainteasers and mathematical games. While for the remainder of the week we were taught, in these sessions with the headmaster we were encouraged to think, to apply logic and deduction, to sharpen our mental faculties. In turn, it afforded the headmaster an opportunity to identify those with real academic potential, rather than simply the ability to learning by rote.

Each week, the pupils were presented with one brainteaser to solve. These puzzles were usually wrapped up in some form of a story, provided as much to obscure as to enlighten, but in essence they were mathematical in nature. However, the most satisfying puzzles offered a logical short cut that allowed one to reach the correct conclusion without resorting to algebra, geometry or quadratic equations. The example of the water lily and the lily pond will illustrate their equal mix of beguiling simplicity and frustrating complexity:

The Water Lily and the Lily Pond
The Squire had recently constructed a lily pond at the Manor House which was 15 feet long and 8 feet wide. He ordered his gardener to plant a water lily in one corner of the pond. The chosen water lily was a newly introduced cultivar renowned for its rapid growth. Indeed, the water lily grew at such a prodigious rate that the surface area of the pond it covered doubled each day. Seven days after it was planted, the water lily was covering half the surface area of the lily pond. How long would it take until the pond was completely covered?

There is an immediately obvious and wrong answer, which is fourteen days. There is also a long-winded mathematical approach, which involves calculating the area of the pond and thus the speed of growth of the water lily, which ultimately yields the correct answer. Much more satisfying is the logical approach, which quickly discerns that the dimensions of the pond are as immaterial as whether the water lily was planted by the gardener or the Squire himself.

If none of the pupils arrived at the correct answer in the course of the thirty-minute lesson, we were left to puzzle until the following week when the answer was revealed. In the same spirit, rather than simply reveal the solution for those who didn’t reach it instantaneously, I’ll leave you with little thinking time. After all, there’s nothing quite as satisfying as labouring over a puzzle until the moment when, quite unexpectedly, the light dawns.