What Digits Solve the Cryptarithm (THINE)/(EVENLY) = 2/15?

[K Sengupta]

Solve this cryptarithm, where each of the letters represent a different decimal digit from 0 to 9. None of T and E is zero, and “EVENLY” is even.

(THINE)/(EVENLY) = 2/15

 

[Jimmy Snyder]

Spoiler

83596/626970 = 2/15

This was found by brute force.
I have been finding these by writing and running a program that runs through all the possible values of the digits and prints out the ones that satisfy the defining equation. As such, they are not very interesting. Is there any method to these puzzles other than brute force that I am missing? In other words, is there any brain teasing going on here?

 

[davee123]

I have been finding these by writing and running a program that runs through all the possible values of the digits and prints out the ones that satisfy the defining equation. As such, they are not very interesting. Is there any method to these puzzles other than brute force that I am missing? In other words, is there any brain teasing going on here?

There's some-- for instance, we know that (THINE) must be divisible by 2, and that (EVENLY) must be divisible by both 5 and 3. Hence, Y must be 0 or 5, and E must be one of 0, 2, 4, 6, 8. Next, you can rule out 8 as a possibility for E, because the maximum value for (THINE) is 98764, which means (EVENLY) has a max of 740730, meaning that E cannot be greater than 7. Next, we can rule out T=1, because the minimum value of (EVENLY) is 101235, which would make (THINE) be 13498. Since E =/= T, T clearly can't be 1.

Anyway, you can plug away at these bit by bit in that fashion, but yeah, it's certainly easy to write a brute force program to do the job.

DaveE

 

[Jimmy Snyder]

There's some-- for instance, we know that (THINE) must be divisible by 2, and that (EVENLY) must be divisible by both 5 and 3. Hence, Y must be 0 or 5, and E must be one of 0, 2, 4, 6, 8. Next, you can rule out 8 as a possibility for E, because the maximum value for (THINE) is 98764, which means (EVENLY) has a max of 740730, meaning that E cannot be greater than 7. Next, we can rule out T=1, because the minimum value of (EVENLY) is 101235, which would make (THINE) be 13498. Since E =/= T, T clearly can't be 1.

Anyway, you can plug away at these bit by bit in that fashion, but yeah, it's certainly easy to write a brute force program to do the job.

DaveE

Can such plugging away actually solve the problem or simply reduce the time (nearly zero) that the inevitable program takes to run?

 

[davee123]

Can such plugging away actually solve the problem or simply reduce the time (nearly zero) that the inevitable program takes to run?

I haven't tried this one all the way through, so I'm not sure if you can solve it totally with the given clues without brute force, but then again, sometimes you can do some clever tricks that are way beyond me. As for making the program run faster, yes. It isn't really necessary for these cases, since exploring all 3628800 possibilities is pretty quick for a computer. But sometimes it makes the difference between your program running for 2 seconds versus 2 weeks.

If you want some difficult brute force problems, try:
http://projecteuler.net/

They start out easy, but as you go on, you can use some elegant math to simplify the solutions and make your brute force algorithms manageable.

DaveE

 

[Edgardo]

Spoiler

Found by brute force:

Found solution 1: t=8 h=3 i=5 n=9 e=6 v=2 l=7 y=0
thine=83596 evenly=626970
thine*15=1253940, evenly*2=1253940