[Guest]

Substitute each of the capital letters in bold by a different base ten digit from 0 to 9 to satisfy this alphametic equation. None of the numbers can contain any leading zero.

SCENE + (IT)*(IS) = NEATLY

Note: While a solution is trivial with the aid of a computer program, show how to derive it without one.

[Pickett]

Substitute each of the capital letters in bold by a different base ten digit from 0 to 9 to satisfy this alphametic equation. None of the numbers can contain any leading zero.

SCENE + (IT)*(IS) = NEATLY

Note: While a solution is trivial with the aid of a computer program, show how to derive it without one.

Unfortunately I have to go to a meeting, so I cannot finish this right now...but I'll give you my first thoughts...This one's a little harder, since there must be 1 digit that isn't used at all...

IT * IS can only be a 3 or 4-digit number (100 up to 9801)...so, we are adding a 5 digit and a 3 or 4 digit to get a 6 digit number...so we know that N must be 1 (since the 6 digit number must start with 1)...and S must be 9 (because adding a 3 or 4 digit number to a 5 digit number to get a 6 digit number means that the 5 digit number must be between 90199 and 99900)...which using the pattern of letters, limits us to numbers between 92313 and 98717

We can also deduce that E must be 0, because if E = 2, then (IT)*(IS) must be a 5 digit number (123456 - 98212 = 25244)...and that can't be. and E can't be 1, because N is 1.

So we now have:

9C010 + (IT)*(I9) = 10ATLY

We now know that the smallest number we can get from (IT)*(IS) is 4335 (because 102345 - 98010 = 4335).

The largest number we can get from (IT)*(IS) is 87*89 = 7743

So our range of values for (IT)*(IS) = 4335 to 7743

Our only possible numbers for SCENE are 92010, 93010, 94010, 95010, 96010, 97010, and 98010.

And our range of values for NEATLY = 102345 to 108765

We can then say that I must be 6, 7, or 8 because anything less would yeild a (IT)*(IS) result less than 4335.

if I=8, then C = 3,4,5,6,7

if I=7, then C = 5,6,8

if I=6, then C = 7,8

Like I said, that's all I've got for now...if no one has solved this when my meeting's over, I'll work on it some more

[Guest]

Unfortunately I have to go to a meeting, so I cannot finish this right now...but I'll give you my first thoughts...This one's a little harder, since there must be 1 digit that isn't used at all...

IT * IS can only be a 3 or 4-digit number (100 up to 9801)...so, we are adding a 5 digit and a 3 or 4 digit to get a 6 digit number...so we know that N must be 1 (since the 6 digit number must start with 1)...and S must be 9 (because adding a 3 or 4 digit number to a 5 digit number to get a 6 digit number means that the 5 digit number must be between 90199 and 99900)...which using the pattern of letters, limits us to numbers between 92313 and 98717

We can also deduce that E must be 0, because if E = 2, then (IT)*(IS) must be a 5 digit number (123456 - 98212 = 25244)...and that can't be. and E can't be 1, because N is 1.

So we now have:

9C010 + (IT)*(I9) = 10ATLY

We now know that the smallest number we can get from (IT)*(IS) is 4335 (because 102345 - 98010 = 4335).

The largest number we can get from (IT)*(IS) is 87*89 = 7743

So our range of values for (IT)*(IS) = 4335 to 7743

Our only possible numbers for SCENE are 92010, 93010, 94010, 95010, 96010, 97010, and 98010.

And our range of values for NEATLY = 102345 to 108765

We can then say that I must be 6, 7, or 8 because anything less would yeild a (IT)*(IS) result less than 4335.

if I=8, then C = 3,4,5,6,7

if I=7, then C = 5,6,8

if I=6, then C = 7,8

Like I said, that's all I've got for now...if no one has solved this when my meeting's over, I'll work on it some more

Picking up where you left off (which is the direction my thinking was headed, as well), since T is on both sides of the equation, and all six digits of the answer have to be different, you end up with: 98010+(63)*(69)=102357. So, S=9, C=8, E=0, N=1, I=6, T=3, A=2, L=5, and Y=7. The unused digit is 4. There may be another solution, but this one works.

Edited June 18, 2009 by bonanova
Please use spoilers when replying with a solution

[Guest]

S can only be 9 as SCENE is 5 digit number and IT*IS wil be 4 digit while NEATLY is 6 digit number

This means that N can only be 1. E could be 0,1,2... however since 1 is already taken, and with Neatly becoming 12ATLY, the difference with SCENE would become 5 digit which can not be achieved with IT*IS

So E must be 0.

So, we have

9C010 + IT*I9 = 10ATLY

Now, the max value of IT*I9 is 87*89 = 7743 and the min value is 29*28 = 812...

The differnce of 102345 (min value of NEATLY) and 98010 (max value of SCENE) = 4335

So, if C = 8, I can be either of 6 or 7

if C = 7, I can be only 8

if C =6, I can be only 8

Now, NEATLY has T as the 100s digit and SCENE has 010 in the end.. this means that IT*I9 must have T as the 100s digit.

This is possible with the following cases

C = 8; I = 6 then T = 3 or 4

C = 7; I = 8; T = 5

C = 6; I = 8; T = 5

For C = 8; I = 7 then there is no value of T

So, there are only 4 cases to check;

and the only combination that satisfies is

C = 8; I =6; T = 3

So, SCENE is 98010

IT is 63

IS is 69

NEATLY is 102357

[Guest]

i'm confuzzled

[Guest]

Substitute each of the capital letters in bold by a different base ten digit from 0 to 9 to satisfy this alphametic equation. None of the numbers can contain any leading zero.

SCENE + (IT)*(IS) = NEATLY

Note: While a solution is trivial with the aid of a computer program, show how to derive it without one.

took some thought. Have to break it down little by little. series of number cannot start with a 0, next you are adding a 5 digit number to a multiplication of 2 two-digit numbers and getting a 6 digit number. highest scenario is it=99 and is=99 which multiplied together does not equal 10000, or the lowest 5 digit number. There for you are adding a 5 digit number with a 4 digit number (at most) and getting a 6 digit number. Again worst case scenario is that SCENE= 99999 and it*is= 9999. Meaning that NEATLY <200,000 and N has to equal 1, and same logic would say that S=9 it would be interesting if somebody solves the whole thing without paper or a calculator.

[Guest]

took some thought. Have to break it down little by little. series of number cannot start with a 0, next you are adding a 5 digit number to a multiplication of 2 two-digit numbers and getting a 6 digit number. highest scenario is it=99 and is=99 which multiplied together does not equal 10000, or the lowest 5 digit number. There for you are adding a 5 digit number with a 4 digit number (at most) and getting a 6 digit number. Again worst case scenario is that SCENE= 99999 and it*is= 9999. Meaning that NEATLY <200,000 and N has to equal 1, and same logic would say that S=9 it would be interesting if somebody solves the whole thing without paper or a calculator.

I got as far as working out that N=1, and S=9. Not sure I could work it out without pencil and paper though!

[Guest]

This sure was fun. I took a lot of the same steps that Pickett explained. It was just the ending half that had me going; finding I and T was probably the only tedious part. I could not find a mathmatical way to produce an answer. But since I only had so many choices for C, it was just a matter of trial and error for finding I and T. I'm new here and wanted to find more problems such as this, anyone know if any others? These kind of questions fill my boring life at my work desk.