# Homework Help: If each letter stands for a number

1. Sep 7, 2013

### Bashyboy

1. The problem statement, all variables and given/known data
If each letter stands for a number from 0 to 9 (and different letters stand for different
numbers), find the numerical value of each letter in this addition:

2. Relevant equations

3. The attempt at a solution
(1) N + D = R

(2) I + I = E ----> I = E/2, which means E must be even

(3) R = E

These were my initial conjectures. However, I now feel that they are wrong. At first, I thought that there was no carrying over of digits--which would be a helpful fact--; then I realized that D was in the 1000's place, which implies that there must have been some carrying over.

Could someone help me?

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Last edited: Sep 7, 2013
2. Sep 7, 2013

### UltrafastPED

Since the problem is in base 10 (digits are 0-9) we can avoid carrying by expanding the numbers thus:
(1000*D +100*E + 10*E + R) = (10*I + N) + (100*R + 10*I + D)

3. Sep 7, 2013

### Bashyboy

So, were my three initial observations correct or incorrect?

4. Sep 7, 2013

### Enigman

3) is definitely incorrect........

5. Sep 7, 2013

### Bashyboy

What do you mean by "start with D?" Yeah, I figured that 3) would be wrong, seeing as they don't meet the requirement of being distinct numbers.

6. Sep 7, 2013

### UltrafastPED

The problem is a puzzle - it resolves to 999D + 110E = 99R + 20I + N where D,E,R,I,N are five distinct digits.

At this point you can try different combinations for a while ... a small computer program would find any possible solutions very quickly. Of course there may be something very clever that can be done, but I prefer algorithms to cleverness - they get the work done.

7. Sep 7, 2013

### Bashyboy

Now, can I take the equation 999D + 110E = 99R + 20I + N, solve for N, and choose values for D, E, R and I such that N is an integer and N does not equal one of the other letters?

8. Sep 7, 2013

### UltrafastPED

Yes ... though you actually need to provide a guess for all five at one time since there is only one equation!

If you are clever you may find a way to generate more equations ... but I don't; that's why I suggested a small computer program to cycle through all of the possible combinations. In other words, guess and check!

Looking at the original expressions it is clear that there is a carry from the I+I column - therefor D=R+1. That gives you another equation to work with. And to get a carry from 2I (an even number) you must have I>4 (because a carry from the previous column can add no more than 1). We also know that D is the result of a carry, hence D=1.

So I guess that cleverness can simplify things after all - all I needed was a bite to eat and I began to see things.

Now you can work through the rest ... and when you cannot go any further, guess the rest.

9. Sep 7, 2013

### Bashyboy

How is it clear that there is a carry over from the I+I column?

10. Sep 7, 2013

### UltrafastPED

Otherwise R=E which violates the rules.

11. Sep 7, 2013

### UltrafastPED

Since D is the result of a carry, D=1; but DE = R + 1 (from the carry from I + I) ... thus R=9, D=1, and E=0.

Thus DEER = 1009.

I doubt that my method is good enough for the Putnam exam - but I do keep plugging away! I hope that you are having as much fun as I am. I now notice that the first column has N+D=R or maybe N+D=1R if there is a carry ... but D=1, R=9 ... so N=8.

That means there is no carry so I + I = 1E =10, so I = 5.

So 58 + 951 = 1009. So it can be solved by paying attention - thus proving that computers are not always required.

12. Sep 7, 2013

### Bashyboy

I am rather confused. First, you write DE = R + 1. Secondly, what is the difference between N + D = R and
N + D = 1R?

13. Sep 7, 2013

### Bashyboy

Also, why isn't it R = E + 1, instead of DE = R + 1

14. Sep 7, 2013

### Bashyboy

UltrafastPED, I found out where you are wrong. E can't equal 10.

15. Sep 7, 2013

### UltrafastPED

Well the rest is up to you!

16. Sep 7, 2013

### Bashyboy

All right. Let's recap. D must be one, because it is the result of a carry over, so no other letter can assume the value 1. E must be even, because 2I = E; however, we know that $E \ne I \ne 0$, otherwise I = E, which doesn't meet the conditions specified in the problem. If we solve for N, then we get the function N(E,R,I) = 999 + 110E -99R -20I. Before I proceed with substituting in random values, how many combinations do we have? I know there are 4 choices for E. Does that leave 8 choices for R, and 7 choices for I?

Oh wait!!!!!!! E can't exceeed 10, meaning we can't choose any numbers exceeding 5 for I. This means that I can be 2, 3, or 4.

Let me work out the rest of the details. Feel free to respond before I do!

17. Sep 7, 2013

### Bashyboy

Hmmm, this doesn't seem to be working. If I = 2, then E = 4. So, are function is N(4,R,I). Let R = 9, then

N(4,9,4) = 508, which isn't an integer between 0 and 9.

18. Sep 7, 2013

### Bashyboy

I think I might have to ask my professor concerning this problem.

19. Sep 7, 2013

### Enigman

What is the maximum carry over when two numbers less than 10 are added?
From this you can get both R, D and E.

20. Sep 7, 2013

### OmCheeto

You forgot I.

:tongue:

21. Sep 7, 2013

### Enigman

The O.P. needs to figure something out himself...

22. Sep 7, 2013

### OmCheeto

From my notes from this morning, I was only able to initially logically deduce the values of R & D in my brain. Your brain must be bigger than mine. hmmm.......

And why is this post in "Calculus & Beyond Homework" section? This belongs in either the "Fun, Photos & Games" or "Precalculus Mathematics Homework" section. I'd push that gosh darned report button, but it never says anything about misplaced questions.......

oh poop. They changed the wording.....

I didn't get the memo.........

:grumpy:

--------------------
* always changing the rules, and never informing the flotsam........

Last edited: Sep 8, 2013
23. Sep 8, 2013

### Bashyboy

What precisely do you mean by this? Have I not been able to determine so portions of the problem myself?

24. Sep 8, 2013

### Bashyboy

The maximum carry over is one, which is a fact I used in my solutions.

25. Sep 8, 2013

### ehild

You know already that D=1, R=9 and E=0. Substitute the letters with these numbers in the sum.

......I N
...9 I 1
--------
1 0 0 9

The sum of two one-digit numbers can not exceed 9+9=18. Therefore N+1 = 9. N?
As there is no carry at N+1, I+I=0 or I+I=10. But I cannot be 0. I+I carries 1. So what is I?

ehild