> BDA5@ &bjbj22 "@XX,400000000}$RZy00yy00GGGy:00}Gy}G"Gii0$hti}0i`q`i`i0Grl000$1123-234-345-456-567-678-789
Assuming an ordered set of numbers, where each corresponding digit is one more (when comparing to the previous number) or one less (when comparing to the following number):
Ordered Set: {123, 234, 345, 456, 567, 678, 789}
For instance, in the numbers 123 and 234, the difference for the first digit for both these numbers is a value of 1 (1 vs 2). For the second digit, the difference for both these numbers is a value of 1 (2 vs 3). For the third digit, the difference for both these numbers is a value of 1 (3 vs 4).
If you take the differences between the adjacent numbers in this ordered set, the result is 111. This makes sense because each digit is a value of 1 away from a corresponding digit from an adjacent number in the set. Since you add 1 to each digit, this means that you are adding one ones place, one tens place and one hundreds place. Putting these together you get
1 + 10 + 100 = 111.
Below, the first column is the difference (between adjacent numbers in the set), while the next two columns are operations equivalent to the first column, where the difference is split in half (these next two are an alternate approach to doing differences). For instance, 789 678 can be rewritten as 789 700 which gives you the first half of the difference, and 700 678 which gives you the second half of the difference. Adding these sub-results together gives you the same value as the original difference 789 678.
234 123 = (234 - 200) + (200 - 123) = 34 + 77 = 111
345 234 = (345 - 300) + (300 - 234) = 45 + 66 = 111
456 345 = (456 - 400) + (400 - 345) = 56 + 55 = 111
567 456 = (567 - 500) + (500 - 456) = 67 + 44 = 111
678 567 = (678 - 600) + (600 - 567) = 78 + 33 = 111
789 678 = (789 - 700) + (700 - 678) = 89 + 22 = 111
Looking at the sums (i.e.: 34 + 77), place the first number of each sum in an ordered set and call it Top Number Set, place the second number of each sum in an ordered set and call it Bottom Number Set (the first number is the difference between numbers of higher value top, while the second number is the difference between numbers of lower value bottom). This is the result:
Top Number Set: {34, 45, 56, 67, 78, 89}
Bottom Number Set: {77, 66, 55, 44, 33, 22}
Notice that Top Number Set is increasing, while Bottom Number Set is decreasing. This is because of the constraint that when corresponding numbers in each set are summed they must equal the number 111. Thus, going from 34 to 45 is an increase of 11, and going from 77 to 66 is a decrease of 11. Since this change in value is constant, as one number goes up and the other number goes down, the sum remains the same.
If you assume that each number represents a clock time (i.e.: 123 is 1:23), then the numbers 567, 678 and 789 would be left out since they could not correspond to a standard clock time.
New Clock-time Ordered Set: {1:23, 2:34, 3:45, 4:56}
The properties discussed above for the Ordered Set would still apply to this New Clock-time Ordered Set. However, because these numbers are now clock times, differences will be performed assuming that there are 60 minutes for every hour.
2:34 1:23 = (2:34 2:00) + (2:00 1:23) = 0:34 + 0:37 = 71 minutes
3:45 2:34 = (3:45 3:00) + (3:00 2:34) = 0:45 + 0:26 = 71 minutes
4:56 3:45 = (4:56 4:00) + (4:00 3:45) = 0:56 + 0:15 = 71 minutes
Notice that for above, the differences between adjacent clock times is constant at 71 minutes. Stated another way, this is 60 minutes + 11 minutes, or 1 hour 11 minutes (which looks a lot like the value 111 for the Ordered Set).
These properties can now be extended to differences of adjacent numbers in permutation sets. In other words, new sets can be created with numbers similar to the Ordered Set, where each number is a permutation of a corresponding number in the Ordered Set. For instance, based on Ordered Set = {123, 234, 345, 456, 567, 678, 789}, you now have Permutation #1 Ordered Set = {132, 243, 354, 465, 576, 687, 798}. It turns out that the difference between adjacent numbers in this Permutation #1 Ordered Set is a constant value of 111, as before. However, for this to work, the pattern that was used to generate each permutation has to be the same for every number in the permutation set. For instance, the first number in the Ordered Set is 123. The first number in the Permutation #1 Ordered Set is 132. The pattern that was used to generate 132 was to keep the first digit 1 and to swap the next two digits, so 23 becomes 32, and 132 is generated. This same pattern is applied to the rest of the numbers generated in the Permutation #1 Ordered Set.
Ordered Set value => Permutation #1 Ordered Set value:
123 => 132
234 => 243
345 => 354
456 => 465
567 => 576
678 => 687
789 => 798
In addition, each number in the Ordered Set has 6 permutations. This allows us to generate a total of 6 Permutation sets: 1 Ordered Set and 5 Ordered Permutation Sets.
(The underlined numbers are not eligible to be clock-time numbers)
Permutation #1: {123, 234, 345, 456, 567, 678, 789} [Ordered Set]
Permutation #2: {132, 243, 354, 465, 576, 687, 798}
Permutation #3: {213, 324, 435, 546, 657, 768, 879}
Permutation #4: {231, 342, 453, 564, 675, 786, 897}
Permutation #5: {312, 423, 534, 645, 756, 867, 978}
Permutation #6: {321, 432, 543, 654, 765, 876, 987}
Because the patterns used to generate each Permutation Set are the same for each of these sets, the difference between a number in one Permutation set and a corresponding number in another Permutation set will be the same across all numbers in the sets. For instance, between Permutation #6 and Permutation #1, the difference between the first number in each set is 321 123 = 198. Now we can use this result and add it to each number in Permutation #1 to generate a matching corresponding number in Permutation #6. 234 (the second number in Permutation #1) + 198 = 432 (the second number in Permutation #6. 345 (the third number in Permutation #1) + 198 = 543 (the third number in Permutation #6), and so on.
In this way, we can determine the differences between Permutation Sets.
Permutations Set Difference Combinations (pairs in brackets are repeats):
1-2, 1-3, 1-4, 1-5, 1-6
[2-1], 2-3, 2-4, 2-5, 2-6
[3-1], [3-2], 3-4, 3-5, 3-6
[4-1], [4-2], [4-3], 4-5, 4-6
[5-1], [5-2], [5-3], [5-4], 5-6
[6-1], [6-2], [6-3], [6-4], [6-5]
Permutation Set Differences, using the first number of each set to do the differences, always taking the larger minus the smaller number:
1-2: 132 123 = 9
1-3: 213 123 = 90
1-4: 231 123 = 108
1-5: 312 123 = 189
1-6: 321 123 = 198
2-3: 213 132 = 81
2-4: 231 132 = 99
2-5: 312 132 = 180
2-6: 321 132 = 189
3-4: 231 213 = 18
3-5: 312 213 = 99
3-6: 321 213 = 108
4-5: 312 231 = 81
4-6: 321 231 = 90
5-6: 321 312 = 9
Ordering these differences ascending:
1-2 / 5-6: 9
3-4: 18
2-3 / 4-5: 81
1-3 / 4-6: 90
2-4 / 3-5: 99
1-4 / 3-6: 108
2-5: 180
1-5 / 2-6: 189
1-6: 198
Putting the differences above into a matrix:
9, 90, 108, 189, 198
9, 81, 99, 180, 189
90, 81, 18, 99, 108
108, 99, 18, 81, 90
189, 180, 99, 81, 9
198, 189, 108, 90, 9
There appears to be a relationship between the differences of permutation sets that are shown in this pseudo-symmetric matrix, but this relationship is difficult to ascertain. However, one obvious relationship is that each of these numbers is divisible by the number 9. If you divide each of these numbers by 9, you get a new matrix.
1, 10, 12, 21, 22
1, 9, 11, 20, 21
10, 9, 2, 11, 12
12, 11, 2, 9, 10
21, 20, 11, 9, 1
22, 21, 12, 10, 1
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