Can you replace the letters with 10 consecutive primes such that the sum of numbers on each line is equal? I expect this to be solved with a computer.
First, this image shows examples of translation that preserves the summed groups. As mentioned in comment, there are $12$ equivalent arrangements in this class.
Here is a solution set:
There are seven other solution sets with primes less than 100,000 and countless more with larger primes.