The number of magic squares of order 6

nms6.txt

  17 753 889 189 701 384 304

This result is consistent with stochastic estimates 1.7745(16)·10¹⁹[1] and 1.775392(12)·10¹⁹[2].

Using hundreds of GPUs at cloud resource rental services, it took about six months to complete the counting, Though the number and models of GPUs used at a moment were varying from time to time, the total computation time amounts to about 80,000 hours of GeForce RTX-4090.

Because of the extraordinary volume of the calculation, it is not easy to deny a possibility the result is contaminated by accidental errors. I have done moderate cross-check trying to exclude such errors and am in the course of a full cross-check. I would appreciate confirmations or disputes by others.

1. K. Pinn and C. Wieczerkowski, Number of Magic Squares From Parallel Tempering Monte Calrlo, International Journal of Modern Physics C, 9 April 1998.
2. W. Trump, Estimate of the number of magic squares of order 6.

Since the number is too huge to count in one task, the whole task is divided into many small sub-tasks. Counts for the sub-tasks are available.

• Definition of subsets
• List of 4,329 subtotals
• List of subsubtotals (234MBytes .gz)

Strategies in counting magic squares

CUDA code

• Counts magic squares of an order 3..6 up to M-transformations.
• Runs at a typical speed of 2.5G counts/sec on Nvidia GeForce RTX-4090 for order 6.
• Nvidia GPU of Pascal architecture (sm_60) or newer is assumed.
• Multi GPU systems are supported. Be cautious, however, when you use a multi-4090 system .
• Compiling and linking: nvcc -O3 -arch=sm_60 -maxrregcount=40 -Wno-deprecated-declarations ms.cu -lcrypto
  • If you don't need md5 checksums, add -DnoMD5 and drop -Wno-deprecated-declarations and -lcrypto.
• For orders less than 6, specify the order by a compiler option -DN=order.
• The executable takes 0, 2, or 4 parameters. The 1st and the 3rd parameters are just place holders.
  • ./a.out
    counts all magic squares
  • ./a.out dummy representative_magic_series_in_hex
    counts magic squares whose representative magic series is equal to a given hex number.
  • ./a.out dummy1 representative_magic_series_in_hex dummy2 2nd_largest_magic_series_in_hex
    counts magic squares whose representative magic series and the 2nd magic series parallel to the representative are as specified.
  • The code doesn't check the validity of parameters given by users. Invalid parameters will result in a wrong answer or a runtime error.

Non-CUDA code in C using pthread

• Compiling and linking: gcc -O3 -DNTH=number_of_threads ms.c -lpthread -lcrypto
• Options -DnoMD5 and -DN=order have the same effects as in the Cuda code.
• Much slower than the CUDA code, but easier to read.

• the number: file (Don't copy-paste, download instead.) timestamp
• the CUDA code: file timestamp
• the pthread code: file timestamp
• verification site

2023/07/18
— Hidetoshi Mino, Ph.D. Professor emeritus, University of Yamanashi, Japan