The number of magic squares of order 6

Computing resources available in clouds are diverse in their performance, price and lifetime. To save cost, I had to continuously seek inexpensive resources and create instances in daily basis. The total number of instances ever created exceeded five hundreds. I did the following as a minimal verification of the resouces:

• Every time a new instance created a benchmark program which counts a small subset was executed and the time and the result was recorded. If the answer was wrong, that resource was never used. I actually saw a few such resources and it is noteworthy that some versions of docker environment with multiple RTX-4090s produced wrong answers due to a failure of inter-GPU atomic transactions. I avoided multiple 4090s working together and used them separately instead.
• As a postmortem verification, independent re-counting have been done for every instance at a sampling rate of once per day. If any wrong results were found, all results that resource produced were considered unreliable.

If every subtotal is calculated twice and the two results match, the counts should be considered correct (provided the code is correct). The re-counting is in progress and is done up to 6.5% as of 2023/07/15.