Here is a solution with minimal side-sum:
This was found by brute-force.
The code below will traverse between the bounds as identified by trolley813 and prints a solution with each side-sum (42 & 54 take a while but the rest are much faster). The output solutions are in row-major order.
from itertools import combinations, permutations
There is no limit to this! The Green Tao theorem tells you that the sequence of prime numbers contains arbitrarily long arithmetic progressions. This means that using $a+b\cdot n$ you can get as many primes as you want for some $a,b$ and consecutive values of $n$.
But the theorem does not tell you how to find $a,b$. The longest known such sequence can be ...
The function with rule
produces distinct primes for $n$ up to $25$.
For proof, see the third bullet point on this list of prime number records. It is valid for $x=0,1,...,23$, so I substitute $n=x+2$ so that the set of valid inputs begins at $2$. The function is clearly strictly increasing and so the primes must be distinct.
It is apparently the longest ...
Instead of a cartesian plane, let's consider instead a...
That means rotating $n$, $m$ and $k$ around, so that
So armed with this nomenclature, let's run a few iterations starting with only the $(0,0)$ point:
...we can conclude that the number of moves needed to clear the...
I'm sure I remember a recent puzzle based on the same idea, but I'm failing to find it. There was
Aha, found it:
I'm not sure whether this should be regarded as a duplicate of that one. It isn't the same question but it's clearly closely related.
Based on my previous answer of a similar question, I created a Python program to brute-force all the possible solutions.
To calculate the number of possible solutions:
That is a big number, but not too big for a computer with some time.
However, there is two caveats:
Here is the code:
from dataclasses import dataclass
from enum import Enum
from typing ...
The medians have values ranging from 2 to 8, so exactly one of these values does not appear as the median of one of the 6 rows/columns.
Here are three templates that allow you to choose any missing median value.
For completeness, here are the solutions this produces:
Partial answer with quick upper and lower bounds for the range.
I believe the highest number you can get should be:
The lowest should be:
Which means that at most there could be:
That said, the problem states numbers, which means we need to include decimals.
Heading out shortly, but thoughts about an answer that probably won't actually help:
My best solution so far is
This is done with the following thermometer
Here are three other thermometers with the same score:
If you find any more thermometers that have this score and are not symmetric to any of the three in this post I would be happy to see them.
Symmetry here means any combination of:
Reversing the ...