Math Games - Number Sequences (part 2 of 3)

In this second article about 'What comes next?' number sequences, we'll consider how to solve more complex arithmetic sequences, in which math skills becomes more significant than an ability to simply recognise and repeat patterns.

nintendo ds reading games

More Complex Arithmetic Sequences
In math, an arithmetic sequence or progression is a sequence of numbers in which the difference between any two successive numbers in the sequence is the same. In the concluding example of the previous article, we tested whether a sequence of numbers was a simple arithmetic progression by comparing the numerical difference between adjacent pairs of numbers. It is important that adjacent pairs of numbers are chosen, as more complex forms of arithmetic progression may be misidentified as simple arithmetic sequences if non-adjacent pairs are selected.

READING GAMES

Consider the sequence of numbers:

2 6 4 8 6 10 _ _

If you worked out the mathematical difference between the first pair and last pair of numbers, in this case both +4, you might be tempted to conclude that you were dealing with a simple arithmetic sequence with a common difference of +4. However, the difference between the first pair of numbers and the second pair of numbers is +4 and -2 respectively, so clearly there is something more complicated happening. In such cases, the most reliable course of action is to work out the difference between each successive pair of numbers. In this case, the differences work out to be:

+4, -2, +4, -2, +4,

At the simplest level, you could say the numerical series is generated by adding four to the initial number to make the second number, then subtracting four from the second number to make the third, and so on ad infinitum. This approach will work and allow you to complete 'What comes next?' number series generated in this manner. However, there is certain mathematical inelegance to the approach. To a mathematician, the series comprises two simple arithmetic progressions, each with a common difference of two, which have been interleaved so that so that the series takes, alternately, numbers from one progression and then the next.

The two series when separated are:

2 4 6 8

6 8 10 12

and the next two numbers in the series 8 and 12.

Whilst in this example there are few real advantages to separating out the simple arithmetic progressions. More complex numerical series may alternate three or more arithmetic progressions, which makes determining what math principles underlie the numerical sequence increasingly difficult.

As we're ready to move on in the next article to look at geometric progressions and unique number series, we'll conclude with a little brain teaser.

What two letters come next in the following sequence?

O T T F F S _ _

This is an interesting example of the 'what comes next?' question, in that arriving at the correct answer will, for most people, involve elements of math, pattern matching and the spark of inspiration which we tend to consider innate intelligence. The question is also as likely to be answered correctly and in a reasonable length of time by an intelligent 5 year old as by an intelligent adult, which makes it an ideal IQ test question for measuring innate intelligence. So what is a reasonable length of time in which to arrive at the correct answer? As part of an IQ test, answering correctly in less than 10 seconds would put you in genius territory, while taking around a minute is perhaps the average. For anyone who is really struggling with the question, please be assured that an answer and an explanation will be forthcoming in the final 'number series' article.

Math Games - Number Sequences (part 2 of 3)

Hannah McCarthy is Marketing Executive for Education City, a leading supplier of e-Learning software in the US. With over 300 activities in the math module, Education City provides interactive math games and smartboard lessons to enhance whole class math teaching, together with fun math games for kids to enjoy at home.

READING GAMES