First, write them down on paper in order from small to large. Naturally, some figures will be repeated, so write them under the original figures. After writing 50 groups, I found that some numbers are not repeated and can be discarded, because they can't fit into the nine-square grid (the number we are looking for must have at least two sets of Pythagoras numbers to fit into the nine-square grid, so that Pythagoras numbers are formed vertically and horizontally).
After the first step of screening, we further found that not every number with two sets of pythagorean numbers meets the requirements, such as 5, whose pythagorean numbers are 5, 12, 13 and 3, 4, 5, but obviously 3, 4, 5 can't be filled in, because 3 has only one set of pythagorean numbers! So numbers like 5 must also be discarded. The method is to check the Pythagorean number of each group. If one number in a set of numbers has only one Pythagorean number, then this set of numbers is discarded.
The above work seems tedious, but if you copy these 50 sets of Pythagoras numbers into word and use the search function, it will be completed soon!
After two rounds of screening, we only got 12 sets of Pythagoras numbers! They are:
i= 15 j=20 k=25
i= 15 j=36 k=39
i=20 j=48 k=52
i=24 j=32 k=40
i=24 j=45 k=5 1
i=25 j=60 k=65
i=32 j=60 k=68
i=36 j=48 k=60
i=39 j=52 k=65
i=40 j=75 k=85
i=45 j=60 k=75
i=5 1 j=68 k=85
It is easy to find that they all meet our current requirements, except that 60 has four sets of Pythagoras numbers, and all other numbers have two sets of Pythagoras numbers. Although we have made great progress, the next question is how to fill the Pythagorean numbers of 17 into nine squares. It doesn't take much effort to get it. I began to fill a corner of the nine squares. The first number is 15. Then the first line has two filling methods, namely 15, 20, 25 and 15, 25, 20 (the two filling methods are essentially the same, which will be found later). Correspondingly, the first column also has two filling methods, namely15,36,39 and15,39,36. It looks much clearer here. According to the number of 12 groups I provided, it is easy to complete the nine squares. There are at least nine filling methods, and the exchange between rows and columns is irrelevant. The following is one of the situations:
15 25 20
36 60 48
39 65 52
Don't forget, I have provided the number 17, and 60 seems to have a good relationship. It divides the number 17 into two groups, the eight numbers in the previous example and the remaining eight numbers. It is easy to find that the remaining 8 numbers can be filled into 9 squares like 60. The following is one of the situations:
24 32 40
45 60 75
5 1 68 85
So there are finally 18 filling methods.