My September 9, 2016, blog post ("Next Number" Puzzles) pointed out the meaninglessness of the typical "next-number" puzzle. It did this by showing that two such puzzles in the STICKELERS column by Terry Stickels had more than one solution. In addition to the solution proposed in the column, another was found in a polynomial that interpolated the given members of the sequence. Of course, the very nature of the question "What is the next number?" is absurd because the next number could be anything. At best, such puzzles should require finding a pattern for the given sequence, admitting that there need not be a unique pattern.
The STICKELERS column continued to publish additional "next-number" puzzles, now no longer of interest. However, the remarkable puzzle of December 30, 2017, caused me to pull from the debris on my retirement desk the puzzle of July 15, 2017, a puzzle I had relegated to the accumulating dust thereon.
The members of the given sequence appear across the top of the following table that reproduces the graphic used to provide the solution.
It turns out that the pattern in the graphic can be expressed as 100 – (-1)k k(k+1)/2, k=0,…, a pattern Maple helped find. By the techniques in my earlier blog, an alternate pattern is expressed by the polynomial
which interpolates the nodes (1, 100), (2, 101) ... so that f(8) = -992.
The most recent puzzle consists of the sequence members 0, 1, 8, 11, 69, 88; the next number is given as 96 because these are strobogrammatic numbers, numbers that read the same upside down. Wow! A sequence with apparently no mathematical structure! Is the pattern unique? Well, it yields to the polynomial
which can also be expressed as
Hence, g(x) is an integer for any nonnegative integer x, and g(6) = -401, definitely not a strobogrammatic number. However, I do have a faint recollection that one of Terry's "next-number" puzzles had a pattern that did not yield to interpolation. Unfortunately, the dust on my desk has not yielded it up.