Stuck on a difficult problem? Try freewheeling through it

Attacking a problem from a variety of angles can often lead to a sudden insights - also known as the aha moment

Photograph: iStock

Photograph: iStock

 

Many problems in mathematics that appear difficult to solve turn out to be remarkably simple when looked at from a new perspective. George Pólya, a Hungarian-born mathematician, wrote a popular book, How to Solve It, in which he discussed the benefits of attacking problems from a variety of angles. Pólya considered all kinds of problems, not just mathematical ones. The book was translated into several languages, and more than a million copies were sold. It is still in print.

Let us look at a few examples of lateral thinking in maths. Suppose we have 27 players in a knock-out competition. How many matches in total will there be in the tournament? We start with a first round of 13 matches, one player getting a bye into the second round. With 14 players left, seven matches in the second round leaves seven players. Three quarter-final matches, with one player getting a bye, leaves four. Three more matches – two semi-finals and a final – yield the winner.

Adding the matches, we get a total of 26. Is all this cumbersome arithmetic really necessary? No. Simple logic gives the answer in a flash: with 27 players there are 26 losers. Each match determines a loser, so 26 matches are required. Voilà.

Deliberately disruptions

The technique of lateral thinking was popularised by the Maltese psychologist and inventor Edward de Bono who, in his books, gave many examples of problem-solving. One key idea is deliberately to disrupt routine reasoning processes and allow the mind to freewheel randomly. Thus, hidden connections and unexpected relationships are discovered. Sudden insights, often called aha moments, may result.

I recall a standard lamp in the drawing room of my home that cast a beautiful light-curve on the wall. Intrigued by its elegant form, I spent an hour or so calculating the shape of the boundary curve and eventually arrived at the equation of a hyperbola, one of the family of curves known as conic sections. Suddenly, the light dawned on me. The lampshade restricted the light to a cone with a vertical axis, and the wall cut through the cone, defining the hyperbola. I did not need the hour of algebra: the answer was evident immediately from simple geometry.

As a final example, consider two men, initially six miles apart, walking towards each other at three miles per hour. A bee flies at 15mph from the nose of one to the nose of the other and back again, repeatedly zig-zagging to and fro until the two men meet. How far does the bee fly? We can add up all the segments of the bee’s oscillatory expedition and calculate the sum of a convergent infinite series to obtain the answer. But there is a much simpler way: the men meet after one hour, so the bee flies 15 miles. By a change of perspective, the problem becomes trivial to solve.

The flight-of-the-bumblebee problem was posed to John von Neumann, another renowned Hungarian-born mathematician. Von Neumann thought briefly and announced the answer. His inquisitor remarked: “Oh, I see you’ve spotted the quick way to do it”. “What quick way?” responded von Neumann. He had summed the infinite series in his head.

Puzzlewright Press has recently published a small book of lateral-thinking puzzles, written by Prof Des MacHale of UCC and Paul Slone (see lateralpuzzles.com).

  • Peter Lynch is emeritus professor at the school of mathematics and statistics, University College Dublin. He blogs at thatsmaths.com
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