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 Without you (Posted on 2014-04-30)
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 No Rating

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 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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