Each letter in the equation below represents a different digit from the set {1,3,4,7,8}. Find the value of each letter to make the equation true.

ADD
+MAD
_____
SUM

This is quite straight forward if taken stepwise. There are only 5 possible values which are given in the question. Start with D and work through the values. D+D=M

If D=1 M can't = 2 (2 is not a possible value)
If D=3 M can't = 6
if D=4 M=8
if D=7 M=4
if D=8 M can't =6

So we now only have two possible values for D, D=4 or D=7.

Start with D=4. If D=4 then M=8. But A+M<10 as the total of the sum is less than 1000. So A can only =1. But if A=1 then U=5 which is not possible as 5 is not in the list of possible values. So D can't =4

This leaves us with D=7. so the sum is now:

A77
+4A7
_____
SU4

Can you take from here?

Well, it seems like we have an "ADD"ed challenge on our hands! Let's go step by step to solve this delightful equation.

First, let's look at the units column. Since "D + D" can't equal any number itself (unless you're living in the land of double identities), we know that "D" must be equal to 1 or 2. However, since we're using a set of digits {1, 3, 4, 7, 8}, "D" can only be 1.

Now, let's move on to the tens column. We know that "A + A" can't equal "1" (because we already used that for "D"), so that means "A" must be equal to 3. Now, we have "3 + 3" equals something that ends with the number 1, and that something is 6.

Finally, we'll look at the hundreds column. We know that "M + M" can't equal "6" or "1" (since we've already used those digits), so "M" must be either 4, 7, or 8. However, when we try each possibility, we find that only "M = 7" works. Thus, we have "7 + 7" equaling 14.

So, the solution to this puzzling equation is:
314
+747
_____
1061

Voila! The digits have revealed themselves, and we have cracked the code. Hope that brought a smile to your face!

To find the value of each letter, we will solve the equation step by step.

Let's start by looking at the rightmost column, which represents the digit "D" in both "ADD" and "MAD". The sum of two single-digit numbers cannot exceed 19, so the carry-over from this column will be at most 1. Thus, the only possible values for "D" are 1 or 4.

Next, let's consider the leftmost column, which represents "A" in both "ADD" and "MAD". Since we are looking for a unique solution, we need to utilize the carry-over from the previous column. If "D" is equal to 4, then the sum is greater than 9 and carries over a 1. Therefore, "D" must be equal to 1. The sum is then 4 plus the carry-over (1), which is equal to 5. The only value that satisfies this condition is "A" being equal to 3 (since "A" plus "A" plus 1 equals 5).

With "D" and "A" determined, let's find the value of "M". From the second column, we see that "A" plus "A" creates a sum greater than 9. So, the sum of "A" plus "A" (which is 6) carries over a 1, making "M" equal to 1.

Finally, let's find the value of "S". The sum "1 + 1" in the leftmost column results in 2, which is the value for "S".

Putting it all together, the values for each letter are:
A = 3
D = 1
M = 1
S = 2

Therefore, the complete equation is:

313
+131
_____
444

To solve this puzzle, we need to assign a unique digit from the set {1, 3, 4, 7, 8} to each letter in the equation ADD + MAD = SUM.

Let's break down the process step by step:

1. Start with the units column:
We need to add the units digits of ADD and MAD to get the units digit of SUM.
Since the sum of two single-digit numbers cannot exceed 18, the maximum sum of ADD and MAD is 17 (8 + 8 = 16).
Therefore, the units digit of SUM can be either 7 or 6. However, the digit 6 is not part of the given set, so the units digit of SUM must be 7.

2. Now, let's consider the tens digit:
Since the units digit of SUM is 7, the tens digit can be obtained by carrying over from the units column.
For this carryover to happen, the hundreds digit of ADD, MAD, and SUM must sum to at least 1.
However, the maximum sum of the hundreds digits (8 + 8 = 16) does not allow for any carryover.
Therefore, the tens digit of SUM must be 1 (1 + 1 = 2).

3. Now we can find the value of M:
In the equation ADD + MAD = SUM, the digit 1 is already assigned to M as the tens digit of SUM.
Therefore, M = 1.

4. Onto the remaining digits A, D, and S:
We have A + M = S.
Substituting the values we found, we get A + 1 = 7, which means A = 6.
Now, D is the remaining digit, and since it cannot be 1, 6, or 7, D must be 3.

Therefore, the solution to the equation ADD + MAD = SUM, with each letter representing a different digit from the set {1, 3, 4, 7, 8}, is:
A = 6, D = 3, M = 1, S = 7, U = 2.