Hello everyone this is my first question on here.
What is the relation between a and b when a-b=c | a,b,c ∈ Z | 123 ≤ a,b,c ≤ 987 and a,b,c have distinct digits 1-9.
For example one possible value of a and b are: a = 459, b = 173, c = a-b = 286. This works as no digit repeats, it is a three digit number, 0 is not used and a is greater than b.
I've gone through every possibility with a computer which gives 336 possible combinations of a and b. Obviously though a-b=c and a-c=b are both true so every combination has a "duplicate".
I have not yet found a way of deriving these numbers with a mathematical rule, as it doesn't fit with any standard formulas. If anyone could find one i'd be very interested.
Here's the list of possible values of a (left-column) and b if it can be of aid:
- 459 : 173, 176, 183, 186, 273, 276, 283, 286
- 468 : 173, 175, 193, 195, 273, 275, 293, 295
- 486 : 127, 129, 157, 159, 327, 329, 357, 359
- 495 : 127, 128, 167, 168, 327, 328, 367, 368
- 549 : 162, 167, 182, 187, 362, 367, 382, 387
- 567 : 128, 129, 138, 139, 218, 219, 248, 249, 318, 319, 348, 349, 428, 429, 438, 439
- 576 : 182, 184, 192, 194, 382, 384, 392, 394
- 594 : 216, 218, 276, 278, 316, 318, 376, 378
- 639 : 152, 157, 182, 187, 452, 457, 482, 487
- 648 : 251, 257, 291, 297, 351, 357, 391, 397
- 657 : 218, 219, 238, 239, 418, 419, 438, 439
- 675 : 182, 183, 192, 193, 281, 284, 291, 294, 381, 384, 391, 394, 482, 483, 492, 493
- 693 : 215, 218, 275, 278, 415, 418, 475, 478
- 729 : 143, 146, 183, 186, 543, 546, 583, 586
- 738 : 142, 146, 192, 196, 542, 546, 592, 596
- 783 : 124, 129, 154, 159, 214, 219, 264, 269, 514, 519, 564, 569, 624, 629, 654, 659
- 792 : 134, 138, 154, 158, 634, 638, 654, 658
- 819 : 243, 246, 273, 276, 352, 357, 362, 367, 452, 457, 462, 467, 543, 546, 573, 576
- 837 : 142, 145, 192, 195, 241, 246, 291, 296, 541, 546, 591, 596, 642, 645, 692, 695
- 846 : 317, 319, 327, 329, 517, 519, 527, 529
- 864 : 125, 129, 135, 139, 271, 273, 291, 293, 571, 573, 591, 593, 725, 729, 735, 739
- 873 : 214, 219, 254, 259, 614, 619, 654, 659
- 891 : 234, 237, 254, 257, 324, 327, 364, 367, 524, 527, 564, 567, 634, 637, 654, 657
- 918 : 243, 245, 273, 275, 342, 346, 372, 376, 542, 546, 572, 576, 643, 645, 673, 675
- 927 : 341, 346, 381, 386, 541, 546, 581, 586
- 936 : 152, 154, 182, 184, 752, 754, 782, 784
- 945 : 162, 163, 182, 183, 317, 318, 327, 328, 617, 618, 627, 628, 762, 763, 782, 783
- 954 : 216, 218, 236, 238, 271, 273, 281, 283, 671, 673, 681, 683, 716, 718, 736, 738
- 963 : 215, 218, 245, 248, 715, 718, 745, 748
- 972 : 314, 318, 354, 358, 614, 618, 654, 658
- 981 : 235, 236, 245, 246, 324, 327, 354, 357, 624, 627, 654, 657, 735, 736, 745, 746
Also, I've noticed the sum of every "a" value is 18, this may be significant.
A sub question that I have aswell is why the number of possibilities is 336. I noticed it is equal to $8*7*6$ but why I don’t know.
Thankyou for taking the time to read this question.