Sequence Puzzles

Given a sequence consisting of the whole numbers 1, 4, 9, 16, 25, 36, and 49, what number comes next in the sequence?

The most likely answer is 64–the next number in a sequence of squares of consecutive integers, starting with 1.

Such sequence puzzles are a staple of textbook exercises, brainteaser collections, and various intelligence and aptitude tests. Some sequences are easy to figure out, some have multiple interpretations, and others can require considerable head-scratching before the pattern becomes evident.

Neil A.J. Sloane of AT&T Shannon Labs in Florham Park, N.J., has been collecting number sequences ever since he was a graduate student at Cornell University in the 1960s. He described nearly 6,000 examples in his 1995 book The Encyclopedia of Integer Sequences and has added many thousands of additional examples to an online edition of the book (see http://www.research.att.com/~njas/sequences/).

One useful feature of Sloane’s online compendium is the ability to enter a set of numbers and search for information about that sequence. For example, suppose you enter the numbers 1, 2, 3, 6, 11, 23, 47, 106, 235. Among other things, the results page tells you that the next term is 551, that this sequence is associated with trees having n nodes, and that there is a formula for calculating the sequence’s terms.

As a way to demonstrate his online encyclopedia, Sloane recently assembled an entertaining page of sequence puzzles, available at http://www.research.att.com/~njas/doc/OEIS1.pdf. It starts off with several simple classic sequences (perfect squares, Fibonacci numbers, and so on), then quickly moves into more perplexing territory.

For example, what do you make of the sequence 1, 2, 3, 7, 43, 1807, 3263443? It turns out that the terms are sometimes called Euclid numbers and the sequence’s next member is 10650056950807.

Sloane has additional puzzles at http://www.research.att.com/~njas/sequences/Spuzzle.html. When you give up, you can just click on a link to see the answer. That’s a handy shortcut you don’t have available to you when you’re taking your SATs.