# [0-9] digit puzzle

Sorry for the long delay between publishing articles. My illness has been catching up to me, and I’ve had a rough couple of weeks.

## Puzzle

Here’s a fun little number puzzle that is easy to solve with a few lines of brute force code, but with a little thinking, and a pad and paper, can be solved pretty much by hand and some common sense.

Rearrange the digits [0-9] inclusive to make three numbers:

*x,y,z*(No leading zeroes). These numbers should match the equality that x+y=z. Find all the solutions.*Use the digits [0-9] to make three numbers: x,y,z so that x+y=z*

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## Thinking

The first thing to think about is the number of significant digits. When you add two numbers together, at a minimum, you have to have at least the same number of digits, for the longest, as what you started with; (positive) numbers never get shorter by adding. This means the number of digits in

*z*cannot be less than four. (For example, if*z*were three digits long, then either*x*or*y*would need to be at least four digits long, which makes no sense).With other thinking, we see that

*z*cannot have five (or more) digits. If*z*had five digits, then it would have need to have grown at least one digit (the max length of either*x*or*y*would be four digits, and the only way to grow the length of a number is with a carry). To convert a four-digit number to a five-digit number would require a cascading carry all across the entire number. In this puzzle, the maximum number of cascades that can occur to a four digit number by adding a single number is two, and this is where the second digit is a 9 (the least significant digit of*x*and*y*get added, which causes a carry to the next significant digit, which is a 9, and this can roll over once more, but then things get quenched as the max carry is a one - There are no more ways to propagate the carry). The number of digits in*z*, therefore, must be four.*The number of digits in*

*z*must be four.This gives us only two patterns to consider. With

*z*having four digits, then*x*and*y*are either {2,4} or {3,3} digits long.## {2,4}

Without loss of generality, and to reduce duplicates, I’ll assume

*x*is the two-digit number.Because every digit needs to be distinct, and both

*y*and*z*are four digits, then the addition of*x*needs to cascade across the entire four-digit number so that most significant digit changes (changing every digit along the way). The only way that this can occur is when*y*is in the format*D 9 ? ?*so that the cascading carry rolls over the most significant digit of*z*giving a solution of the form*(D+1) 0 ? ?*From here it’s a simple exercise to permute the digits not used to create a pair of two-digit numbers that carry forward a one, and use distinct digits in their solutions too.Here is a list of all the {2,4} solutions:

26+4987=5013

27+4986=5013

34+5978=6012

34+5987=6021

37+5984=6021

38+5974=6012

43+5978=6021

47+2968=3015

48+2967=3015

48+5973=6021

56+1978=2034

56+1987=2043

57+1986=2043

58+1976=2034

64+2987=3051

65+1978=2043

67+2948=3015

67+2984=3051

68+1975=2043

68+2947=3015

73+5948=6021

74+5938=6012

75+1968=2043

76+1958=2034

78+1956=2034

78+1965=2043

78+5934=6012

78+5943=6021

84+2967=3051

84+5937=6021

86+1957=2043

86+4927=5013

87+1956=2043

87+2964=3051

87+4926=5013

87+5934=6021

27+4986=5013

34+5978=6012

34+5987=6021

37+5984=6021

38+5974=6012

43+5978=6021

47+2968=3015

48+2967=3015

48+5973=6021

56+1978=2034

56+1987=2043

57+1986=2043

58+1976=2034

64+2987=3051

65+1978=2043

67+2948=3015

67+2984=3051

68+1975=2043

68+2947=3015

73+5948=6021

74+5938=6012

75+1968=2043

76+1958=2034

78+1956=2034

78+1965=2043

78+5934=6012

78+5943=6021

84+2967=3051

84+5937=6021

86+1957=2043

86+4927=5013

87+1956=2043

87+2964=3051

87+4926=5013

87+5934=6021

## {3,3}

To convert two three-digit numbers into a four-digit number, there needs to be a carry to the most significant digit. (There are a couple of solutions that simple that simply cause this carry in the most significant digit, but most solutions cascade the carry).

Because of the limits of carry, the most significant digit for

*z*will need to be 1.You will quickly find patterns in pairs of digits, and their chiral solutions.

Here is a list of all the {3,3} solutions:

246+789=1035

249+786=1035

264+789=1053

269+784=1053

284+769=1053

286+749=1035

289+746=1035

289+764=1053

324+765=1089

325+764=1089

342+756=1098

346+752=1098

347+859=1206

349+857=1206

352+746=1098

356+742=1098

357+849=1206

359+847=1206

364+725=1089

365+724=1089

423+675=1098

425+673=1098

426+879=1305

429+876=1305

432+657=1089

437+589=1026

437+652=1089

439+587=1026

452+637=1089

457+632=1089

473+589=1062

473+625=1098

475+623=1098

476+829=1305

479+583=1062

479+826=1305

483+579=1062

487+539=1026

489+537=1026

489+573=1062

537+489=1026

539+487=1026

573+489=1062

579+483=1062

583+479=1062

587+439=1026

589+437=1026

589+473=1062

623+475=1098

624+879=1503

625+473=1098

629+874=1503

632+457=1089

637+452=1089

652+437=1089

657+432=1089

673+425=1098

674+829=1503

675+423=1098

679+824=1503

724+365=1089

725+364=1089

742+356=1098

743+859=1602

746+289=1035

746+352=1098

749+286=1035

749+853=1602

752+346=1098

753+849=1602

756+342=1098

759+843=1602

764+289=1053

764+325=1089

765+324=1089

769+284=1053

784+269=1053

786+249=1035

789+246=1035

789+264=1053

824+679=1503

826+479=1305

829+476=1305

829+674=1503

843+759=1602

847+359=1206

849+357=1206

849+753=1602

853+749=1602

857+349=1206

859+347=1206

859+743=1602

874+629=1503

876+429=1305

879+426=1305

879+624=1503

249+786=1035

264+789=1053

269+784=1053

284+769=1053

286+749=1035

289+746=1035

289+764=1053

324+765=1089

325+764=1089

342+756=1098

346+752=1098

347+859=1206

349+857=1206

352+746=1098

356+742=1098

357+849=1206

359+847=1206

364+725=1089

365+724=1089

423+675=1098

425+673=1098

426+879=1305

429+876=1305

432+657=1089

437+589=1026

437+652=1089

439+587=1026

452+637=1089

457+632=1089

473+589=1062

473+625=1098

475+623=1098

476+829=1305

479+583=1062

479+826=1305

483+579=1062

487+539=1026

489+537=1026

489+573=1062

537+489=1026

539+487=1026

573+489=1062

579+483=1062

583+479=1062

587+439=1026

589+437=1026

589+473=1062

623+475=1098

624+879=1503

625+473=1098

629+874=1503

632+457=1089

637+452=1089

652+437=1089

657+432=1089

673+425=1098

674+829=1503

675+423=1098

679+824=1503

724+365=1089

725+364=1089

742+356=1098

743+859=1602

746+289=1035

746+352=1098

749+286=1035

749+853=1602

752+346=1098

753+849=1602

756+342=1098

759+843=1602

764+289=1053

764+325=1089

765+324=1089

769+284=1053

784+269=1053

786+249=1035

789+246=1035

789+264=1053

824+679=1503

826+479=1305

829+476=1305

829+674=1503

843+759=1602

847+359=1206

849+357=1206

849+753=1602

853+749=1602

857+349=1206

859+347=1206

859+743=1602

874+629=1503

876+429=1305

879+426=1305

879+624=1503