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Without you (Posted on 2014-04-30) Difficulty: 4 of 5
Given:
1. N1=*u*
2. N2 =*u*
3. N1 not equal to N2
4. N1 x N2=u***u*
5. The letter u represents a certain digit.
6. All other digits are replaced by an asterisk (*).
7. No u in the partial products.
8. "x" in statement 4 is a multiplication sign.

Please restore the original numbers N1 and N2.

Bonus question : Eliminating the 7th condition, do we get additional solutions?

See The Solution Submitted by Ady TZIDON    
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Solution computer solution | Comment 1 of 2
If rule 7 is taken strictly, that is without regard to the order of the multiplicands, in neither order do the partial products contain the digit u, there is only one solution:

      874
      874
  -------
     3496
    6118 
   6992 
  -------
   763876
   
With rule 7 taken more loosely there are other solutions, still using rule 7:

 n1  n2  product
 328 626 205328
 329 625 205625
 735 438 321930
 736 437 321632
 873 875 763875
 874 874 763876

For example:

      328    but      626
    x 626    x 328
   ------   ------
     1968     5008   reversing the multiplicands
     656    1252    puts u into partial products
   1968     1878
   ------   ------
   205328   205328
   
Eliminating rule 7 altogether:

 n1  n2  product
 212 518 109816
 212 718 152216
 212 818 173416
 212 918 194616
 214 715 153010
 214 815 174410
 214 915 195810
 215 614 132010
 215 714 153510
 215 814 175010
 215 914 196510
 218 612 133416
 218 712 155216
 218 812 177016
 218 912 198816
 312 418 130416
 312 618 192816
 314 415 130310
 315 414 130410
 315 614 193410
 318 512 162816
 318 612 194616
 320 626 200320
 320 721 230720
 320 826 264320
 320 921 294720
 320 926 296320
 321 625 200625
 321 720 231120
 321 825 264825
 321 920 295320
 321 925 296925
 323 723 233529
 323 727 234821
 323 823 265829
 323 827 267121
 323 923 298129
 323 927 299421
 324 826 267624
 325 621 201825
 325 625 203125
 325 629 204425
 325 721 234325
 325 725 235625
 325 729 236925
 325 821 266825
 325 825 268125
 325 829 269425
 325 921 299325
 326 624 203424
 326 628 204728
 326 720 234720
 326 724 236024
 326 728 237328
 326 820 267320
 326 824 268624
 326 828 269928
 326 920 299920
 327 623 203721
 327 627 205029
 327 723 236421
 327 727 237729
 327 823 269121
 327 827 270429
 328 626 205328
 328 726 238128
 328 826 270928
 329 625 205625
 329 725 238525
 329 825 271425
 412 418 172216
 420 521 218820
 420 621 260820
 421 520 218920
 421 620 261020
 421 625 263125
 423 623 263529
 424 626 265424
 425 621 263925
 425 625 265625
 425 629 267325
 426 620 264120
 426 624 265824
 426 628 267528
 427 623 266021
 427 627 267729
 428 626 267928
 429 625 268125
 431 730 314630
 431 830 357730
 435 738 321030
 435 838 364530
 436 837 364932
 437 736 321632
 438 735 321930
 438 835 365730
 520 521 270920
 520 526 273520
 521 525 273525
 523 523 273529
 523 527 275621
 524 526 275624
 525 525 275625
 525 529 277725
 526 528 277728
 527 527 277729
 530 731 387430
 531 730 387630
 535 738 394830
 536 637 341432
 536 737 395032
 537 636 341532
 537 736 395232
 538 635 341630
 538 735 395430
 540 741 400140
 540 746 402840
 540 846 456840
 541 740 400340
 541 745 403045
 541 845 457145
 543 843 457749
 545 741 403845
 545 841 458345
 546 840 458640
 630 631 397530
 640 641 410240
 641 745 477545
 643 743 477749
 645 741 477945
 646 740 478040
 659 850 560150
 750 751 563250
 750 753 564750
 750 757 567750
 750 759 569250
 751 752 564752
 751 754 566254
 751 756 567756
 751 758 569258
 752 753 566256
 760 861 654360
 761 860 654460
 761 865 658265
 762 864 658368
 763 863 658469
 764 862 658568
 872 876 763872
 873 875 763875
 874 874 763876

As partial products are not a question here, both orders of multiplicand word, so only n1 <= n2 are shown:

 Text1.Text = ""
 For n1 = 100 To 999
   n1s$ = LTrim(Str(n1))
   u$ = Mid(n1s$, 2, 1)
   For a = 1 To 9
   For c = 0 To 9
    n2 = 100 * a + 10 * Val(u$) + c
    prod = n1 * n2
    prods$ = LTrim(Str(prod))
    If Len(prods$) = 6 Then
      If Left(prods$, 1) = u$ And Mid(prods$, 5, 1) = u$ Then
        n2s$ = LTrim(Str(n2))
        If Left(n1s$, 1) <> u$ And Right(n1s$, 1) <> u$ And Left(n2s$, 1) <> u$ And Right(n2s$, 1) <> u$ Then
        If InStr(Mid(prods$, 2, 3) + Right(prods$, 1), u$) = 0 Then
         pp1s$ = LTrim(Str(n1 * c))
         pp2s$ = LTrim(Str(n1 * Val(u$)))
         pp3s$ = LTrim(Str(n1 * a))
         a1 = n1 \ 100: c1 = n1 Mod 10
         tst$ = LTrim(Str(a1 * n2)) + LTrim(Str(Val(u$) * n2)) + LTrim(Str(c1 * n2))
  '       If InStr(tst$, u$) = 0 Then
  '       If InStr(pp1s$ + pp2s$ + pp3s$, u$) = 0 Then
          If n2 >= n1 Then
           Text1.Text = Text1.Text & Str(n1) & Str(n2) & Str(prod) & Chr(13) & Chr(10)
          End If
  '       End If
  '       End If
        End If
        End If
      End If
    End If
   Next
   Next
 Next

Commented lines (with leading apostrophe) were used for the more strict listings, but without the n2 >= n1 stricture.


  Posted by Charlie on 2014-04-30 16:33:40
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