A Number Riddle - Closest Without Going Over
Been thinking about a riddle recently.
What is the largest number still strictly less than 1 that can be described using n mathematical symbols?
To keep it managable I’m thinking about limiting it to basic symbols of arithmetic 1234567890^*/+-.!. In essence the riddle is to create a number closest to 1 without going over. In order to get an idea of whats going on I made a jupyter notebook to brute force and evaluate all the combinations. This pretty quickly failed for two reasons
- floating point errors mean its hard to get that close to 1
- it gets really slow, really fast, like $16^n$
However it worked out the answers for the first few
| n | string |
|---|---|
| 1 | 0 |
| 2 | .9 |
| 3 | .99 |
| 4 | .999 |
| 5 | .9999? |
| 6 | 1-1/9!? |
Unfortuantely the brute force method fails at about the same point at which the analysis starts to get interesting. I think in the n=7 to ~100 range there might be some interesting stuff. On my own I thought of a few strategies which could work:
- raising numbers close to 1 to fractional powers, something like
.9999^.5 - the obvious lower bound, a string of 9s
.99999999… - create the largest number possible, $x$, and
1-1/(x)
Asymptotically the problem may become easier to analyze than some of the large but finite cases. I suspect that some of the medium-range numbers may have ranges where each of these various tricks is the winner, but as n gets larger and larger I think it will eventually be dominated by 1-1/(x) and trying to create the largest number possible in $n-6$ characters.
One possibility there would be 1-1/9!!!!!!!…, although that depends on more of the ‘spirit’ of the question in how you exactly define your syntax. I’d regard that as illegal because of the convention that ! and !! (the double factorial) are different operations, not repeated factorials.
Once again though as $n \to \infty$ I don’t think any of these quibbles will matter. Even requring redundant parenthesis or being pedantic about the syntax would only introduce a constant or maybe polynomial number of additional symbols, which might make the large values of $n$ reduce to a more limited version of Who Can Name the Biggest Number?.