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 has been collecting number sequences ever since he was a graduate student at Cornell University in the 1960s (see the Quanta Magazine article "The Connoisseur of Number Sequences" and the Numberphile video "What Number Comes Next?").
Sloane described nearly 6,000 examples in his 1995 book The Encyclopedia of Integer Sequences and has added thousands upon thousands of additional examples to an online extension of the book, The On-Line Encyclopedia of Integer Sequences (OEIS).
One useful feature of Sloane's OEIS 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 had a practice of assembling entertaining pages of sequence puzzles. One set 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 members of a variant of Sylvester's sequence, and the sequence's next member is 10650056950807.
At the OEIS puzzle page, 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 SAT's.
Originally posted May 19, 2003
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