Home Puzzle mathematics – Counting puzzle #1: Function combinations

mathematics – Counting puzzle #1: Function combinations

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mathematics – Counting puzzle #1: Function combinations

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Not in conjunction with my function optimization puzzles, also sorry for the extremely difficult discrete mathematics puzzle


So as you may or may not know, I have recently uploaded 2 function optimization puzzles which were only part of the inspiration for this puzzle. The rest of the inspiration mostly came from this comment from @AxiomaticSystem:

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Text version for people who aren’t able to load images:

[1 vote] If you want to make more of these, I’d like to see more functions at once – that would allow things like parity to come into play and make the solutions less straightforward. Additionally, I’m curious to see at what point lower limits like the -12 become obsolete. – AxiomaticSystem [14 hours ago] 🖉

I know this was supposed to refer to my minimum function optimization puzzles, but I want to try something here before doing that. Here is the puzzle I have created here:

Let$$\begin{align}f(n):=&\,n^2\\g(n):=&\,2n\\h(n):=&\,4n+1\end{align}$$and let set $\mathbb S$ be the set of pos. integers that can be achieved by applying any combination of $f$s, $g$s, and $h$s to 0. How many pos. integers $\le2048$ are in $\mathbb S$?

Things to mention


  1. Partial answers are 100% allowed and are going to be encouraged on this question! This is because this is a really difficult problem and I don’t want to discourage anyone from answering.
  2. This took me around 1.5 hours to solve{1}, so I have been able to verify that there is an answer that can be reached for this puzzle.
  3. To get the $\color{green}✓$, you have to show your work on how you got all of the numbers and the final answer.

Hint:

This is also somewhat based off of Blackpenredpen’s modification of a 2020 Oxford math admissions question, so this screenshot may come in handy when coming up with a strategy to solve it (from a deleted question of mine on Mathematics.SE that was asking if my solution to the aforementioned problem was correct):

enter image description here


{1}without a brute force algorithm (I can’t program that well), although I think those are allowed under the rules that I have stated as long as every number in $\mathbb S$ is printed and the alg. is given, although please try to solve it yourself first.

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