Solving Sequence Puzzles

Sequence puzzles can sometimes be solved just by looking at the puzzle. For instance if you are given 1,2,3,4 then it is obvious the next term is 5.

Here is an example where the solution is not quite so obvious:

4, 9, 16, 25

What is the next number in the sequence above? If you can't spot straight away what the sequence is, one useful method is to look at the differences between the individual terms of the puzzle. So the difference between 4 and 9, 9 and 16 and so forth to see if any pattern or regularity appears there.

Here, we realise that the differences are as follows: 5, 7, 9. The next level of difference is then 2, 2 - not needed here as the pattern has now become obvious, but sometimes useful.

So if we see a pattern at this level, what does it tell us about the sequence and how can it help us to solve the sequence? Well, here there is now an obvious pattern so we simply add on 11 to find the next term - the answer is 36, then 13 for the next term 49, and so on. The sequence is solved.

Let's look at another example:

1, 8, 81, 1024

Here the differences are: 7, 73 and 943. There seems to be no steady pattern, but we can clearly see that the numbers are increasing by a larger number each time. This suggests to us that some sort of factor is involved with this sequence. The best way to solve these puzzles is to look at ways to get from the first to the second term where the totals are still low, and then extrapolate on any patterns and see if they result in the same answer.

So we could try 1 cubed, 2 cubed to get 1 and then 8. However, this would give 27 as the answer for term 3 - but we notice this is an order lower, so 3 to the four would give the result. This gives us an idea as to the answer.

The answer seems to be that the exponent part is increasing by one each time as well as the number, so we end up with 1 squared, 2 cubed, 3 to the 4, 4 to the 5, making the next number in the sequence 5 to the power of 6, which is 15625.